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744 is the nineteenth number of the form where , and represent distinct
prime numbers (
2,
3, and
31; respectively).[1]
It can be represented as the sum of nonconsecutive
factorials,[2] as the sum of four consecutive primes ,[3] and as the product of
sums of divisors of consecutive
integers;[4] respectively:[i]
The number partitions of the
square of
seven (
49) into prime parts is 744,[17] as is the number of partitions of
48 into at most four distinct parts.[18][iii]
744 is an
abundant number,[23] with an abundance of
432.[24][iv] It is
semiperfect, since it is equal to the sum of a subset of its divisors (e.g., 1 + 2 + 4 + 24 + 62 + 93 + 124 + 186 + 248).[33][v]
Of these
240totatives, 110 are strictly composite totatives that nearly match the
sequence of
composite numbers up to 744 that are
congruent to , which is the same congruence that all prime numbers greater than
3 hold.[88][xii] Only seven numbers present in this sequence are not totatives of 744 (less-than); they are 713, 589, 527, 403, 341, 217, and 155; all of which are divisible by the eleventh prime number
31.[xiii][xiv][xv]
The remaining 130 totatives are
1 and all the primes between
5 and 743 except for 31 (all prime numbers less than 744 that are not part of its
prime factorization) where its largest prime totative of
743[xvi] has a prime index of
132 (the smallest
digit-reassembly number in decimal).[127][xvii] On the other hand, only three numbers hold a totient of 744; they are
1119,
1492, and 2238.[5][xviii]
744 is the sixth number whose totient value has a sum-of-divisors equal to : .[131] Otherwise, the
aliquot sum of 744, which represents the sum of all divisors of 744 aside from itself, is
1176[14] which is the forty-eighth
triangular number,[6] and the
binomial coefficient present inside the forty-ninth row of
Pascal's triangle.[132][xix] In total, only seven numbers have sums of divisors equal to 744; they are: 240,[xx] 350, 366, 368, 575, 671, and 743.[32][xxi] When only the fourteen
proper divisors of 744 are considered, then the sum generated by these is
1175, whose six divisors contain an arithmetic mean of 248,[11][xxii] the third (or fourteenth) largest
divisor of 744. Only one number has an aliquot sum that is 744, it is
456.[14][xxiii]
Graph theory
The number of
Euler tours (or Eulerian cycles) of the
complete, undirected graph on six
vertices and fifteen
edges is 744.[164] On seven vertices, there are 129,976,320 Euler tours. These can only be generated on complete graphs with at least three vertices; the number of tours for three, four and five vertices are respectively 2, 2 and
264 (the latter is the second digit-reassembly number in base ten).[127] On the other hand, the number of Euler tours of the complete
digraph, or directed graph, on four vertices is
256, while on five vertices it is 972,000 (and 247,669,456,896 on six vertices), by the
BEST theorem.[165]
Regarding the largest prime totative of 744, there are (aside from the sets that are the union of all such solutions),
743 independent vertex sets in a four-dimensional 16-vertex
hypercube graph,[167] and
743
connectedcubic graphs with 16 vertices and
girth 4.[168][169] (where the sum of all previous indexes through the seventh is 130, the total number of prime totatives of 744, including 1).[170]
Thrice 743 is 2229,[xxiv] whose average of divisors is 744 (as with thrice any prime number , the average of divisors will be ).[10][11][xxv] This value is a difference of
1110 from 3339, which is the sum of seven hundred and forty-two (742)[xxvi] repeating digits of the
reciprocal of 743, as the forty-eighth
full repetend prime in
decimal[25] (with the smallest number to have a
Euler totient of 744 being 1119).[5]
For
open trails of lengths eight and nine, starting and ending at fixed distinct vertices in the complete undirected graph on five labeled vertices, the number is 132 (the prime index of 743, half 264),[195] that also represents the number of
irreducible trees with fifteen vertices.[196][xxvii] While for the complete undirected graph there are 264 directed Eulerian circuits,[201][202] it is more specifically the number of circuits of length ten in the complete undirected graph on five labeled vertices, and as such it is the twenty-fifth element in a triangle of length on labeled vertices.[203]
Otherwise,
745 is the number of
disconnected simple labeled graphs
covering six vertices, where the most symmetric of these graphs has three pairs of distinct vertices each covered by one edge alone and all three edges
intersecting; this yields the disconnected covering graph on vertices labelled through in a hexagonal arrangement, with the remaining 744 graphs representing all other possible conformations.[204]
456 (the only number to have an aliquot sum of 744) is the number of unlabeled non-mating graphs with seven vertices (where a mating or graph is a graph where no two vertices have the same set of
neighbors), equivalently the number of unlabeled graphs with seven vertices and at least one endpoint;[205] as well as the number of
cliques in the 7-
triangular graph, where every subset of two distinct vertices in a clique are
adjacent.[206] The number of even graphs with seven vertices, where a graph is odd if there is an orientation of its edges and an automorphism that reverses the sense of an odd number of its edges, and even otherwise, is 456.[207][208][xxviii]
In particular, 456 is the aliquot sum of 264, the only number to have this value for [14][xxix]
Specifically, all three common prime factors that divide the Euler totient, sum-of-divisors, and reduced totient of are the smallest and only primes that divide the orders of all twenty-six sporadic groups, in contrast with only six groups whose orders can be divided by the largest consecutive supersingular prime and largest prime factor of seven hundred and forty-four, ;[235]: pp.244–246 [β] three of these belong inside the small family of six
pariah groups that are not
subquotients of [245] The largest
supersingular prime that divides the
order of is [246][247] which is the eighth self-convolution of Fibonacci numbers, where is the twelfth.[226][γ]
The largest three Heegner numbers with also give rise to almost integers of the form which involve . In increasing
orders of approximation,[263]: p.20–23 [δ]
Square-free positive integers over the negated imaginary quadratic field with class number of also produce almost integers for values of , where for instance there is [266][267][ε][ζ]
D4 and F4
is
theta series coefficient of four-dimensional
cubic lattice[280][281] On the other hand, in the theta series of the four-dimensional
body-centered cubic lattice — whose geometry with defines
Hurwitz quaternions of even and odd square
norm as realized in the –cell honeycomb that is dual to the –cell honeycomb (and, as a union of two self-dual tesseractic –cell honeycombs) — the sixteenth coefficient is the seven hundred and forty-fourth coefficient in the series;[282] with coefficient index the forty-ninth non-zero norm.[283][η]
Within finite simple groups
of Lie type,
exceptional Lie algebra holds a minimal
faithful representation in two hundred and forty-eight dimensions, where divides thrice over.[286][287]: p.4 John McKay noted an intersection between finite simple groups of Lie type and those that are sporadic, where symmetries of nodes in the
Dynkin diagrams of complex Lie algebra as well as those of and respectively coincide with the three largest
conjugacy classes of ; where also the corresponding
McKay–Thompson series of sporadic
Thompson group holds coefficients representative of its faithful dimensional representation (also minimal at )[288][237] whose values themselves embed irreducible representation of .[289]: p.6 In turn, exceptional Lie algebra is shown to have a graded dimension [290] whose
character lends to a
direct sum equivalent to,[289]: p.7, 9–11
Osiris or digit-reassembly numbers – equal to the sum of all permutations of subsets of their own digits (e.g.
132 in
decimal vis-à-vis
symmetric groupS6)
Notes
Higher arithmetic
^744 is equal to the sum between the 41st and 44th
indexed prime numbers, inclusive. Indices 15 and 16 of σ(n) multiply to 240, which is the
Euler totient value of 744,[5] and add to 31 which is the largest prime that is not a
totative of 744 (less than). A sum between
24 and
31 generates the tenth
triangular number55, where the eleventh triangular number is 66.[6] 121 is the sum between these two triangular numbers, which is equivalent to the
square of 11. The prime index of 31 and its
permutable prime in
decimal (13)[7] form the third pair of
twin primes(11, 13),[8] whose sum is 24, with respective prime indices 5 and 6[9] that add to 11.
^120 is also equal to the sum of the first fifteen integers, or fifteenth
triangular numberΣ15 n = 1n,[6] while it is also the smallest number with sixteen divisors,[12] with 744 the thirty-first such number.[13] This value is also equal to the sum of all the prime numbers less than 31 that are not
factors of 744 except for 5, and including 1. Inclusive of 5, this sum is equal to 125 = 53, which is the second number after
32 to have an
aliquot sum of 31.[14] In the
Collatz conjecture, 744 and 120 both require fifteen steps to reach
5, before cycling through {16, 8, 4, 2, 1} in five steps.[15][16] Otherwise, they both require nineteen steps to reach 2, which is the middle node in the {1,4,2,1,4...} elementary trajectory for 1 when cycling back to itself, or twenty steps to reach 1.
^744 is the 181st abundant number. Specifically, 432 (twice
216, the
cube of
6) has an aliquot sum of
808,[14] where
433 is the thirty-first full repetend prime in
decimal that repeats 432 digits, which collectively add to 1944.[25] Less than 1000, the sixtieth and largest such prime number is
983, which repeats 982 digits that sum to 4419, a two-digit dual permutation of the digits of 1944 (and where 2475 = 4419 − 1944 has a
reduced totient of 60).[26] 432 is also the twenty-third
interprime between
twin primes,[27] and the eighth such number whose adjacent prime numbers have prime indices that add to a prime number (167 = 83 + 84, the thirty-ninth prime);[9]348, the composite index of 432,[28] is itself the sixth interprime whose adjacent prime numbers have indices that add to a prime number (139 = 69 + 70, the thirty-fourth prime).[9] Importantly, 432 = 33 + 43 + 53 + 63[29] is an
Achilles number, a
powerful number (the thirty-fourth)[30] that is itself not a
perfect power, like
864 (the eleventh index, that is twice its value), and
288 (the totient of 864),[5] as well as
108 (a fourth of 432), and1944.[31] (432 is the sixth member, where the sixth
triangular number is
21,[6] itself the index of 1944 in the list of Achilles numbers. Notice also that 1944 = 1200 + 744, where 1200 = 456 + 744, or in other words σ(456).)[32] On the other hand, 181, the index of 744 as an abundant number, is the sixteenth full repetend prime in base ten,[25] and the composite index of
232,[28] that is the sum of the first five interprimes to lie in-between twin primes with prime sums from respective indices (i.e., 138, 72, 12, 6, 4), as with 432 and 348.[27]109 is the tenth full repetend prime, repeating 108 digits whose sum of repeating digits is
486, a fourth of 1944, where 487 is the thirty-third full repetend prime (and 811 the forty-ninth, one less than the sum of the 180 repeating digits of 181; with 288 = 180 + 108).[25]
^744 is the 183rd semiperfect number, an index value whose sum-of-divisors is 248,[32] and an arithmetic mean of divisors equal to 62 = 31 × 2,[10][11] both of which are divisors of 744 (fourteenth and tenth largest, respectively); otherwise, the sum of its two divisors greater than 1 is 3 + 61 = 64 = 82. 183 is also the eighthperfect totient number,[34] and the number of
semiorders on four labeled elements.[35] It is the largest number of interior regions formed by fourteen intersecting circles, which is equivalent to the number of points in the
projective plane over the
finite field, since 132 + 13 + 1 = 183.[36] It is the number of toothpicks in the toothpick sequence after eighteen stages.[37] Following, 181, 183 is the sixty-second
Löschian number of the form x2 + xy + y2, as a product of the third and twenty-fourth such numbers (
3,
61), of prime indices two and eighteen.[38][9] Furthermore, the smallest number to have exactly four solutions to this
quadratic polynomial is the first taxicab number1729, from (a,b) integer pairs (3, 40), (8, 37), (15, 32) and (23, 25),[39][40] where these four pairs collectively generate a sum of 183. The smallest such number with only two solutions, on the other hand, is 49;[41] wherein 1729 is the 97th such number expressible in two, or more, ways.[42] The probability that any odd number is a Löschian number is 0.75, while the probability that it is an even number is a remaining 0.25.[43] There are precisely 816 integers (uninclusively, equal to the sum-of-divisors of 737, the largest composite totative of 744)[32] between the semiperfect index of 744 and 1000, a number that in-turn has a sum-of-divisors of 2232 = 744 × 3,[32] the 24th
decagonal number,[44] as well as the 30th number that can be expressed as the difference of the squares of consecutive primes in just one distinct way;[45] other members in this sequence include:
888 is the 13th member, the 26th repdigit in decimal[46] equal to the sum between 432 and 456, and between 144 and 744 (where 432 and 456 both have totient values of 144);[5] and equal to the product between the twelfth prime number 37, and 24. The first four members (5, 16, 24, 48) generate a sum of
93, which is the eleventh largest divisor of 744, following 62.[45] 93 represents the number of different
cyclicGilbreath permutations on 11 elements,[47] and consequently there are ninety-three different
realperiodic points of order 11 on the
Mandelbrot set.[48]
Otherwise, 1632 = 1176 + 456, in equivalence with the aliquot sum of 744 (1176)[14] and 456, is the sixteenth number that can be expressed as the difference of the
squares of primes in just two distinct ways, consecutive or otherwise (432, 1728, 1920 and 2232 are also in this sequence).[49] 183 is also the first non-trivial 62-
gonal number.
^Meanwhile, where 279 is the 58th interprime to lie between consecutive odd numbers that are not necessarily twin primes (here at a difference of four), the 58th palindromic number in septenary is the equivalent of
456 in decimal[53] (that is the only number to have an aliquot sum of 744)[14] and represented as 12217, whose digits are the dual permutation of the digits of 744 as the 64th palindrome in base-7 (2112).[53] Explicitly, every positive integer is at most the sum of two hundred and seventy-nineeighth powers (
Waring's problem), preceded by a maximum number of {1, 4, 9, 19, 37, 73, 143}n-powers;[60] where, as aforementioned, 279 is the decimal representation of the ones' complement of 744 in binary. In this sequence, the first six members generate a sum equivalent to the seventh member
143, the composite index of 186[28] that is the radical of 744,[19] as well as equal to the sum of seven consecutive primes starting from 11 through the eleventh prime number: 11 + 13 + 17 + 19 + 23 + 29 + 31. It is also the product between the third
twin prime pair (11 × 13);[8] while figuring in 34 + 44 + 54 + 64 = 74 − 143, which is the second (or third) exception in the sequence of polynomials that starts with 32 + 42 = 52 and 33 + 43 + 53 = 63, after 31 = 41 − 1 (and, 0 = 30 − 1). While in the beginning of the sequence of full repetend primes (with
primitive root 10) the two smallest such numbers are
7 and
17, with the repeating digits of the reciprocal of 7 adding to 27 = 33 that is the seventeenth composite number,[28] there is also 983 + 17 = 1000, and 2 × 279 + 432 = 7 + 983. The span of full repetend primes between 17 and 983 is of fifty-nine integers, where
59 is the seventeenth prime number, and seventh super-prime.[9][61]
^744 is the four hundred and sixth indexed pernicious number, where 406 is the twenty-eighth triangular number;[6] in its base-two representation, the
digit positions of
zeroes are in 1:1:3 or 3:1:1 ratio with the positions of ones, which are in 1:3:1 ratio. Its
ones' complement is 1000101112, equivalent to 279 = 32 × 31 in
decimal, which represents the sum of
GCDs of parts in all partitions of 16 = 42.[55] It is also the number of partitions of 62 = 2 × 31 (a divisor of 744) as well as
63 into factorial parts (without including 0!),[56] and the number of integer partitions of
44 whose length is equal to the
LCM of all parts[57] (with 63 the forty-fourth
composite number,[28] where 44 is itself the number of
derangements of 5,[58] and 63 + 44 =
107 the twenty-eighth
prime number).[9]211210 is the number of repeating decimal digits of 2113 as a full repetend prime in decimal,[25] where 2112 is the 65th interprime to lie between consecutive twin primes,[27] otherwise it is the 317th member between consecutive odd primes,[59] where 317 is the 66th indexed prime number (with 90 and 91 the 65th and 66th composites, respectively, where 181 = 90 + 91).[9][28]279, on the other hand, is the 58th number to lie between consecutive odd prime numbers (
277,
281),[59] with 81 = 92 the fifty-eighth composite number.[28][vi] More specifically, eighty-one is the sum of the repeating digits of the third full repetend prime in decimal,
19,[25] where the sum of these digits is the magic constant of an 18 × 18non-normal yet full
prime reciprocal magic square based on its
reciprocal(1/19).[62][63][64]
^744 is the twenty-third of thirty-one such numbers to have a totient of 240, after
738, and preceding
770. The smallest is
241, the fifty-third prime number and sixteenth
super-prime,[61] and the largest is
1050, which represents the number of parts in all partitions of
29 into distinct parts.[65]
^The value of this sigma function represents the fifteenth sum of non-triangular numbers in-between
triangular numbers; in this instance it is the sum that lies in-between the fifteenth (
120) and sixteenth (
136) triangular numbers[67] (i.e. the sum of 121 + 122 + ... + 135).
^The thirtieth triangular number is 465 = 3 × 5 × 31, equal to the difference between 744 and 279 (equivalently 10111010002 and 1000101112, ones' complement pairs respectively). 30 is the twelfth number m such that 6m + 1 and 6m − 1 are twin primes (
181,
179).[8][69] 465 is also the binomial (31,2) equal to the number of size-2 subsets of {0, 1, ..., 31} that contain no consecutive integers (in light as a triangular number, trailing its sequence by one index).[70][6] In the
Padovan sequence, 465 is the twenty-eighth indexed member equal to the number of compositions of
28 into parts congruent to 2 mod 3, also the number of compositions of 28 into parts that are odd and
greater than or equal to 3, and the number of
maximal cliques in a (28 + 6)–path
complement graph (specifically, in the 34–path).[71]30 is the smallest
sphenic number that is the product of three distinct prime numbers (2, 3, and 5), with the next two such numbers 42 +
66 = 108.[72] Importantly, 30 is the unique point in the sequence of
natural numbers where the ratio of prime numbers to non-primes (up to) is 1/2; when including
0 in +∪ {0} = 0 as a non-prime, this ratio is reached at 29, where a ratio of
one-half is only reached again (here, purely as a ratio between primes and composites) at
37 and
43, which are respectively the twelfth and fourteenth primes (when including 1 as a prime unit, then the thirteenth prime number
41 would be another point of 1:1 ratio, without taking 0 into account).[9] Otherwise, a 1:1 ratio of primes to composites (up to) occurs at two points: 11 and
13, respectively the fifth and sixth prime[9] — whose indices sum to 11, the prime index of 31 — and at
7, when including 0 and 1 as non-primes in , or at
8 when only including 1, and at
9 when considering 0 as non-prime and 1 as a prime unit). Thirty is also the first composite value with three distinct prime factors or more for which the
Möbius function returns −1, as with all numbers with a strictly odd number of prime factors, including prime numbers themselves. Relatedly, 31 is the first number (after
1) to reach a record amplitude between zeros in the associated
Mertens functionM(n), where the next such number is
114,[73] whose arithmetic mean of divisors is 30,[10][11] with a sum-of-divisors of 240.[32] On the other hand, 456 (the only number to have an aliquot sum of 744),[14] 744 and 1176 (the aliquot sum of 744) all set a value of −5 for M(n),[74] respectively the 20th, 57th, and 86th numbers to do so (where 57 is the fortieth composite, and 80 the fifty-seventh),[28] with these indices generating the sum 20 + 57 + 86 = 163, in equivalence with the largest of nine
Heegner numbers. (Also, 20 + 57 = 77, the sum of
base ten digits in the
almost integer approximation of eπ√163 before its decimal expansion.)The sixth pair of numbers to generate a record prime gap (of 14) are the 30th and 31st prime numbers (
113,
127),[75][76][77][9] The previous consecutive primes to generate record gaps are (89, 97), (23, 29), (7, 11), (5, 3) and (3, 2);[78][79] the lesser of these together generate a sum of 127, the thirty-first prime number.
^It is the 168th indexed Zumkeller number, where 168 = 6 × 28 represents the product of the first two
perfect numbers,[81] equal to the fifth Dedekind number,[82] where the previous four indexed members (2, 3, 6, 20) collectively add to 31 — with
496 the thirty-first triangular number,[6] and third perfect number. 168 is also the number of primes below
1000.[9][83] The two sets of divisors of 744 with equal sums are:
(1, 2, 3, 4, 8, 12, 62, 124, 744)
(6, 24, 31, 186, 93, 248, 372)
960 is also the thirty-first Jordan–Pólya number that is the product of factorials 5! × (2!)3,[84] equal to the sum of six consecutive prime numbers 149 + 151 + 157 + 163 + 167 + 173, between the 35th and 40th primes (it is the thirty-fifth such number).[85] The fifteenth and sixteenth triangular numbers generate the sum 120 + 136 =
256 = 28 that is the totient value of 960,[5] and the number of partitions of 29 = 1 + 2 + 3 + 5 + 7 + 11 = 22 + 32 + 42 into distinct parts and odd parts.[86] Like its twin prime 31, 29 is a
primorial prime, which together comprise the third and largest of three known pairs of twin primes (1st, 2nd, and 5th) to be primorial primes,[87] with possibly no larger pairs. In between these lies 30.
^The composite totatives are {25, 35, 49, 55, 65, 77, 85, 91, 95, 115, 119, 121, 125, 133, 143, 145, 161, 169, 175, 185, 187, 203, 205, 209, 215, 221, 235, 245, 247, 253, 259, 265, 275, 287, 289, 295, 299, 301, 305, 319, 323, 325, 329, 335, 343, 355, 361, 365, 371, 377, 385, 391, 395, 407, 413, 415, 425, 427, 437, 445, 451, 455, 469, 473, 475, 481, 485, 493, 497, 505, 511, 515, 517, 529, 533, 535, 539, 545, 551, 553, 559, 565, 575, 581, 583, 595, 605, 611, 623, 625, 629, 635, 637, 649, 655, 665, 667, 671, 679, 685, 689, 695, 697, 703, 707, 715, 721, 725, 731, 737}.Its smallest composite totative is 52 = 25, that is the only number to have an aliquot sum equal to
6,[14] and the second of only two numbers to have a sum-of-divisors of 31 (the largest prime factor of 744), after 16 = 42.[32] Specifically, the twenty-fifth prime number
97 is generated from a sum of 48 and 49. Ninety-seven is the ninth full repetend prime in decimal,[25] whose 96 repeating digits generate a sum of 432, equal to the abundance of 744.[24] 97 is also the first odd prime number that is not a
cluster prime[89] (which precedes the thirty-first prime number 127 in this sequence), where the prime gap for a cluster prime is six or less;[90] the sixth and seventh primes that are not cluster primes are the forty-eighth and forty-ninth primes
223 and
227. 97 is the eleventh prime number p of the form 6m + 1 for a natural number m.[91] Between the median composite totative values of 744, i.e. (413, 415), is
414 — the 333rd composite number[28] — with an
Euler totient of 132[5] that is also the prime index of 743,[9] the largest prime totative of 744. Furthermore, 414 has a sum-of-divisors of 936,[32] the 777th composite number, in-turn the sum-of-divisors of 639 (palindromic with 936), itself the composite index of 777 (also palindromic). Where 414 is the 43rd number to return 0 for the Mertens function, 333 is the 26th,[92] with 43 the fourteenth prime number.[9] Also, the twenty-fifth
repdigit in decimal is 777.[46] Where 25 is the smallest composite totative of 744, the largest such totative is
737, whose aliquot sum is
79,[14] the twenty-second indexed prime number,[9] that is the lesser permutable prime to 97 in decimal.[7]
^713 is the product of the ninth and eleventh prime numbers 23 × 31, where the twenty-third prime number (
83) is the eleventh prime of the form 6m − 1; it is equal to the difference between 744 and 31, with an aliquot sum of 55[14] and totient of
660.[5] ∗589 equal to 19 × 31 is the sum of three consecutive primes (193 + 197 + 199).[93] It is also the ninth
centered tetrahedral number, where 5 and 15 are the first two such numbers (121 = 112 is the fifth).[94] It is the fourteenth third spoke of a hexagonal spiral,[95] and the twentieth quasi-
Carmichael number,[96] with a reduced totient of
90.[26] In the
spt function, 589 is the total number of smallest parts (counted with multiplicity) in all
partitions of 15.[97] ∗527 equal to 17 × 31 is the number of partitions of 31 with equal number of even and odd parts.[98] Its aliquot sum is 49[14] whose arithmetic mean of divisors is 122 =
144, with a sum of its prime factors equal to 48. It is also the maximal number of pieces that can be obtained by cutting an
annulus with 31 cuts,[99] and equivalent to the sum of thirty-one consecutive non-zero integers 2 + 3 + ... + 32; the second-smallest such sum after the third perfect number 496.[81]
^∗403 equal to 13 × 31 is the thirty-seventh number to return 0 for the
Mertens function,[92] and the value of the sum-of-divisors of 144, the first number to generate this value.[32] It is also the first number to hold a totient of
360, of twenty-five such numbers; with 549 the sixth (and the 447th composite number, a value that is itself the 360th composite)[28] and 693 = 144 + 549 the tenth.[5] Also, 403 is the 284th arithmetic number, since its four divisors hold an arithmetic mean of 112 = 7 × 4 × 4, the second such number to hold this value.[10][11]The 112nd prime number is
613,[9] which is the sum of successive composite indexes of 382 and 384 (respectively, 306 and 307),[28] where 383 is the 28th
full repetend prime number in decimal; 613 is also equal to the sum of prime indices of seven consecutive full repretend prime numbers (in decimal): 389, 419, 433, 461, 487, 491, 499, respectively with prime indices 77, 81, 84, 89, 93, 94, and 95[9] (and between the 29th and 35th such primes)[25] — all following 383, where the next full repetend prime after 499 is the 96th prime number
503.[9] 383 is also the second number, after 19, to generate a full
prime reciprocal magic square in decimal whose rows and diagonals generate the same
magic constant (from its repeating digits) of 1719.[62] Within the list of prime numbers, 2017 and 2027 are respectively the 306th and 307th,[9] where the 1719th composite number is 2026.[28]
The arithmetic mean of divisors of 2022, which lies midway between the prime gap of eleven integers (inclusive) just noted, is
507, equal to the sum of prime factors of 2012 = 22 × 503 (inclusive of multiplicities), whose arithmetic mean of divisors in-turn is 588 (half 1176, the aliquot sum of 744) and the second such number to have this value (588), after 1304.[10][11]
1685, on the other hand, is the third number to have an average of divisors that is 507 after 2022, and whose sum-of-divisors (of 1685) is 2028,[32] a number lying between the 307th and 308th prime numbers, 2027 and 2029, the 62nd pair of twin primes);[9][8] 2011 is prime as well, the 305th prime number, with 305 having a composite index 242.[9][28]
1715, the composite index of 2022,[28] is the fourth of thirteen numbers to hold a totient of 1176 = s(744),[5] with an arithmetic mean of divisors equal to
300, which is the twenty-fourth triangular number.[6]
2023, midway between primes 2017 and 2029, has an aliquot sum of 433 (or 432, the abundance of 744, when not counting 1),[14] and a Euler totient of 1632 = 1176 + 456 (the aliquot sum of 744, plus the only number to hold an aliquot sum of 744).[5]
Where 383 repeats 382 digits in base ten, 382 is an arithmetic number too, specifically the first number to have an average of divisors of 144[10][11] while being the seventh number to hold a sum-of-divisors of 576 = 242.[32] Also, 613 + 744 = 1357, alongside 744 − 613 = 131 (both with successive differences of 2 between digits); the latter is the twelfth full repetend prime number in decimal that repeats 130 digits with a sum of 585 = 80 + 81 + 82 + 83, where the 48th full repetend prime, on the other hand, is 743, the largest prime totative of 744 (or 49th in the near-identical sequence of long primes, that instead begins with 2, whose decimal expansion of 1/p has period p − 1).[25][100]Meanwhile, 112 (the 82nd composite) is the composite index of 147 = 48 + 49 + 50[28] with an aliquot sum of 81 = 92[14] and average of divisors of
38.[10][11]147 itself is the composite index of the nineteenth triangular number,
190,[6] equal to the totient and reduced totient of 382.[5][26]744 − 383 =
361 = 192, where 383 is the 76 = 22 × 19 indexed prime number[9] (and
304 fourfold seventy-six, the prime index of 2003, which is the smallest prime number greater than 2000).[9] ∗341 equal to 11 × 31 is the sum between seven consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61), of prime indexes 12 through
18. It is the smallest
Fermat pseudoprime to
binary,[101] and the sixth
centered cube number.[102] It is the number of partitions of 31 such that every part occurs with the same multiplicity,[103] and the thirty-first number in the
Moser–de Bruijn sequence to be a sum of distinct powers of 4 (44 + 43 + 42 + 41 + 40).[104][105] 341 is also the eleventh
octagonal number,[106] equal to the number of
nodes in a regular eleven-sided
undecagon with all diagonals drawn.[107]s(341) = 43,[14] with an arithmetic mean of divisors equal to
96.[11] It is also the tenth
Jacobsthal number,[108] and the total number of largest parts (or equivalently sum of smallest parts) in all partitions of
16.[109]
^∗217 equal to 7 × 31 is the ninth
centered hexagonal number, which also includes as previous members {7, 19, 37, 61, 91, 127, 169}[110] where
127 is the sixth member that is also the thirty-first prime number. 217 is also the sixth non-trivial
dodecagonal number[111] and third non-trivial centered 36-gonal number.[112] It is the third Fermat pseudoprime to
quinary after 124 (a divisor of 744) and 4,[113] and the fifteenth
Blum integer whose distinct prime factors are congruent to 3 mod 4.[114] ✶155 equal to 5 × 31 has an aliquot sum of
37[14] and an arithmetic mean of divisors equal to 48.[5] It is the third number equal to the sum from its lowest prime factor through its largest, the only smaller numbers with the same property are
10 and
39[115] which generate a sum of 49 = 72; the next such number is
371, 1 less than half of 744. There are one hundred and fifty-five total
primitive permutation groups(G, X) of degree 81 that only preserve partitions on the set X by G that are trivial;[116] the sum of the number of all permutation groups of lower degrees is
666, which is
doubly triangular,[117] since it represents the sum of the first thirty-six integers, where
36 is itself the eighth triangular number.[6] The 148 repeating digits of the thirteenth full repetend prime number in decimal
149 (that lies between 109 and 181 in this sequence) [25] sum to 666. The one hundred and fifty-fifth indexed practical number is 744,[50] where the 155th arithmetic number is the forty-eighth prime number
223: the first number to hold an arithmetic mean of divisors of 112, preceding 403.[10][11]
^743 is the twenty-sixth or twenty-seventh lucky number of Euler of the form n2 + n + 41 to yield consecutive prime numbers.[118][119] It is also the twenty-sixth indexed beginning of a maximal chain of primes of the form p(k), ..., p(k+r) for r ≥ 1.[120] It is the forty-eighth
full repetend prime in decimal with
primitive root 10,[25] or similarly the forty-ninth long-period prime when including
2 in this sequence.[100] It is the forty-eighth member in base ten such that every suffix is prime, in which repeatedly removing the most significant digit yields a prime number at every step, until a single-digit prime number remains.[121] Specifically, 743 is the total sum of odd parts in all partitions of 13,[122] and the number of partitions of 31 into partition numbers.[123] Particularly, it is the number of ways to partition 7 labeled elements into sets of sizes of at least 2, and order the sets,[124] and furthermore, the number of
strict integer partitions of
37 with no part divisible by any and all the other parts.[125] Where 743 is the largest prime totative of 744, it represents the sum of all parts of all partitions of all positive integers less than or equal to 25 into an odd number of equal parts or, equivalently, consecutive parts;[126] with twenty-five the smallest composite totative of 744.
^These prime totatives generate a sum of 44,647 (a prime count, the 4641th prime number, an index whose arithmetic mean of divisors is
504)[11] while the composite totatives collectively generate a sum of 44,632 (one less than a prime number, the 4639th prime and 626th super-prime,[61] the latter an index whose value is the fourth and largest number to have an aliquot sum of
316);[14] these sums have a difference of
15, or otherwise collectively have a range of 16 numbers between them, inclusive. On the other hand, differences and sums between the six numbers congruent ±1 mod 6 that are not totatives of 744 are themselves equivalent to divisors of 744, since they are biprimes in proportion with 31 (for example 713 − 589 = 124 and 713 − 527 = 186), while also generating other relevant equalities such as 713 − 217 = 496. The greatest difference between the largest (713) and smallest (155) of these totatives is
558, the seventh number such that the sum of largest prime factors of numbers from 1 to 558 is divisible by 558, the previous and sixth indexed number is 62,[128] the tenth-largest divisor of 744, whose sum-of-divisors is equal to 96;[32] the fifth number in this sequence is
32 (which divides 96). The difference between 558 and 62 is also 496, equal to 713 − 155 − 62; and importantly, 558 is twice 279, which is the ones' complement of 744 in binary in its decimal representation.
^The difference between 2238 and 1492 is 746, where 2238 is twice 1119. Specifically, 746 = 2 × 373 = 15 + 24 + 36 = 2! + 4! + 6! is
nontotient,[20] and equal to the number of non-normal
orthomagic squares with magic constant of 6.[129] Its totient value of 372 is half of 744,[5] that is also 1 less than
373, the 74th indexed prime number[9] that is the smallest difference between numbers that have totients equal to 744 (i.e., 1119 − 1492). 373 is the eleventh two-sided left-and-right
truncatable prime number in decimal, of a total fifteen such primes in base ten.[130] While 373 is the smallest difference between numbers that have a totient value of 744, its digit representation in decimal is the mirror permutation of the digits of 737, the largest composite totative of 744. The sum of these three numbers whose totients equal 744 is 1119 + 1492 + 2238 = 4849.
^1176 is also one of two middle terms in the twelfth row of a Pascal triangle.[133] In the triangle of
Narayana numbers, 1176 appears as the fortieth and forty-second terms in the eighth row,[134] which also includes
336 (the totient of 1176)[5] and
36 (the
square of 6). Inside the triangle of
Lah numbers of the form , 1176 is a member with n = 8 and k = 6.[135] It is a self-
Fibonacci number; the fifty-first indexed member where in its case divides ,[136] and the forty-first 6-
almost prime that is divisible by exactly six primes with
multiplicity.[137]
^240 is the thirty-first quarter square, where 49 is the 14th.[138] Relatedly:[138]
The maximum product of two integers whose sum is 31 is 240.
240 is the sum of the positive integers less than 31 that have the opposite
parity as 31.
240 is the number of nonempty subsets of {1, 2, ..., 31} that contain exactly one odd and one even number.
There are 240 arithmetic progressions of three terms by any means which can be extracted from the set of the first thirty-one natural numbers (starting from 1).
From the starting position in standard chess, 240 is the minimum number of captures by
pawns of the same color to place 31 of them on the same file (column). 240 is also the maximal number of edges that a triangle-free graph of 31 vertices can have.[138]Importantly, 240 is the
polynomial degree of a sixteen-cycle
logistic mapr16,[139][140][141] where 16 is the smallest number to have an aliquot sum of 31.[14]
^The sum of all these seven integers whose sum-of-divisors are in equivalence with 744 is 3313, the 466th prime number[9] and 31st
balanced prime,[142] as the middle member of the 49th triplet of
sexy primes(3307, 3313, 3319);[143] it is respectively the 178th and 179th such prime p where p − 6[144] and p + 6[145] are prime — where 178 is the 132nd composite number,[28] itself the prime index of the largest number (743) to hold a sum-of-divisors of 744.[9][32] 3313 is also the 24th
centered dodecagonal or star number[146] (and 15th that is prime).[147] Divided into two numbers, 3313 is the sum of 1656 and 1657, the latter being the 260th prime number, an index value in-turn that is the first of five numbers to have a sum-of-divisors of 588,[32] which is half of 1176 (the aliquot sum of 744);[14] its arithmetic mean of divisors (of 260), on the other hand, is equal to 49.[10][11] Furthermore, 260 is the average of divisors of 2232 (the fourth largest of five numbers to hold this value), which is thrice 744.
^248 is twice 124 and half
496; the latter which is the third
perfect number[81] — like 6 and 28 (twice 14) — that is also the thirty-first
triangular number,[6] with form 2p - 1(2p − 1) and p = 5 by the
Euclid-Euler theorem. The totient of 248 is 120,[5] with eight divisors that produce an integer arithmetic mean of 60;[11] 496 has a totient of 240 like 744. Where the sum between 744 and the totient 256 = 28 of 960 (the Zumkeller half of 744) is equal to 1000 = 103, their difference is equivalent with 488 = 240 + 248. 248 is also the sum of prime indices of seven consecutivefull repetend prime numbers in decimal: 109, 113, 131, 149, 167, 173, 181,[25] respectively with 29, 30, 32, 35, 39, 40, and 42 prime indices;[9] where 248 is the one hundred and ninety-fourth composite number, and 194 the 149th composite number,[28] the middle term in this consecutive sequence of seven full repetend primes. 149 itself is a strong concatenation of 14 = 7 + 7 and 49 = 7 × 7, joined in the middle by 4, where 7 is the fourth prime number. In the list of successive prime and composite indices of 181, starting with its prime index of
42, the sum generated is 109 = 42 + 28 + 18 + 10 + 5 + 3 + 2 + 1 + 0; with 109 the composite index of 144.[9][28]
^456 is the sixth
icosahedral number,[148] one less than the 88th prime number
457 (itself an index that represents the 64th composite).[9][28] The list of icosahedral numbers includes
124 as the fourth indexed member, where this number is also the third non-unitary stella octangula number after
51 and
14,[149] and the seventhuntouchable number, after 120 and 96;[150] it is the eleventh non-unitary divisor of 744, equal to the sum of all the prime numbers that are not part of its prime factorization, less than 31 (i.e. 5 + 7 + ... + 29). It is furthermore the fourth perfectly partitioned number after 1, 2 and 3.[151] Where 1176 and 456 collectively have a range of seven hundred and twenty-one integers, the difference between
721 and 1000 is 279 (the decimal equivalent of the ones' complement of 744 in binary); where also, 1455 = 1176 + 279 holds an arithmetic mean of divisors that is 294,[10][11] which is one-fourth 1176. This sum also represents the number of
permutation groups of degree 6,[152] and the number of partitions of 48 into distinct parts such that the number of parts is odd,[153] and the number of partitions of 48 into distinct parts such that the number of parts is even.[154] 456 is also, specifically, the number of partitions of the twenty-ninth composite number 44 into prime parts (where on the other hand, 744 the number of partitions of 49 into prime parts).[17] Furthermore, regarding twenty-nine:
the sum of the first ten prime numbers up to 29 yields
129,[155] which is the number of prime totatives of 744 (with the fifth such sum of ten consecutive prime numbers being 264, or twice 132). 129 is also the 97th indexed composite number,[28] where 97 is the 25th prime number,[9]
the twenty-ninth prime number, on the other hand, is
109,[9] which is the number of composite totatives of 744 greater than 25 = 52, its smallest composite totative;
the twenty-ninth record for the sum-of-divisors of a number n is set by σ(744) = 1920.[66]
The count of prime and composite totatives of 744, sans including 1, is 239, which is the 52nd indexed prime number.[9] It is also the sum of prime pairs (23, 29) and (89, 97) that respectively generate consecutive record prime gaps of 6 and 8, which precede the record gap of 14 (the sum of the previous two, which also multiply to 48) between the thirtieth and thirty-first prime numbers (113, 127),[75][76][77][78][79] which sum to 240, the Euler totient of 744.[5] Of highlight, 239 is the only number in concomitance to require a maximum number of
squares (four), a maximum number of
cubes (nine), and a maximum number of
fourth powers (nineteen) to express it[156] — the only other number to require nine cubes is the ninth prime number
23, which is the index of 744 in the list of thirty-one integers to hold a Euler totient value of 240;[5] 23 and 239 are also the third and fourth numbers n (after 1, 2, and 5) that represent the number of partitions of 3n, here 33 and 34 respectively, into powers of 3.[157] When 239 is multiplied with the 628th prime number (4649),[9] it yields the seventh
repunit in decimal (1111111),[156] a number whose arithmetic mean of divisors is 279000 = 279 ×
1000.[10][11] 628 is the 52nd untouchable number, where 52 is the third, following
2 and
5.[150] While the only two positive square
stella octangula numbers are 1 and 9653449 = 31072 = (13 × 239)2,[158] — of indexes 1=12 and 169 = 132 in this sequence[149] — the only solutions to y2 + 1 = 2x4 in
positive integers are (x, y) = (1, 1) and (13, 239).[159] The
fraction239/(132) is the seventh approximation of the
convergingcontinued fraction for √2,[160] where 16
arctan (1/5) − 4 arctan (1/239) is an approximation for πradians.[156]5 × 239 = 1195, the fourth number to have an
arithmetic mean of
divisors equal to 360,[10][11] where specifically, 52 =
13 × 4 =
26 × 2, the prime index of two hundred and thirty-nine, is the fifth
Bell number that counts the number of ways to partition a set of five labeled elements.[161] Three-hundred and sixty, the smallest number with twenty-four divisors[12] and fifty-second
friendly number (the largest known "small" friendly number is 372, or half 744),[162] is the sixth of only seven (
highly composite) numbers such that no number less than twice as much has more divisors.[163] It is also divisible by all integers
less than or equal to ten, except for seven, itself a sum of integers that yields 48.
^In the family of directed
multigraphs that are enriched by the
species of
set partitions, the number of such multigraphs with four labeled directed edges (or arcs) is 2229.[171][172] On the other hand,
2229 is equal to the sum between the sixteenth
Lucas number and its counterpart non-Lucas number, plus 1: (2207, 22).[173][174][175]
2229 is also the number of partitions of 31 with at most 3 odd parts,[176] and the number of partitions of the sixteenth
composite number26[28] that contain a distinct numbers of odd and even parts.[177]
It is also the eighth member n of
polyhexes of size 6n + 1 = 49 that have 6-fold
rotational symmetry, preceding such polyhexes of sizes (1, 1, 2, 4, 11, 37, 136, 540) for 0 ≤ n < 8;[178] for the number of polyhexes of size 31 (n = 6) there are thirty-seven, while for those of size 37 (n = 5) there are 136, which is the sixteenth
triangular number.[6]
2229 is also the number of
endofunctions of
class with distinct
cycle lengths.[179]When grouping successive composite numbers together such that the sum of the –th group is a multiple of the –th prime, 2229 is in equivalence with the twenty-fifth group-sum of composites as divided by twenty-fifth prime number,
97.[180] This sum occurs between 182 composite numbers that span the range (1106, 1313), without including the twenty-six prime numbers in between (starting with 1109 and ending with 1307). The first composite member of the composite grouping is 1106, the fourteenth 14-
gonal number,[181][182] that is also the smallest k such that |M(k)| = 14 (a.k.a. the inverse of the
Mertens function).[183] Where 1106 is the number of regions into which the
plane is divided when drawing twenty-four
ellipses,[184] the number of inequivalentHadamard matrix designs of order 24 is 1106; explicitly, of form 2–(23, 11, 5).[185][186] Also, the first prime number within the sequence of twenty-five consecutive composite numbers that are divisible by the twenty-fifth prime number is
1109, the 186th prime number (an index that represents the thirteenth largest divisor of 744),[9] that is the seventh number such that the Mertens function reaches a new record amplitudes between zeros (of negative 15);[187] the largest prime number within this sequence is the 214th prime, 1307.[9] 1106 also has a sum-of-divisors of 1920,[32] and the arithmetic mean of its divisors is 240,[10][11] respectively the sum-of-divisors and totient of 744.[5] In the list of successive composite indexes — A(n + 1) = A(n)–th composite with A(1) = 47 (cf.
A059408) — 1106 is part of the sequence (47, 66, 91, 122, 160, 207, 264, 332, 413, ..., 1106, ...); a sequence rooted in the fifteenth prime number,
47.[9] The
square of 13 is 169, and the square of 14 is
196; the latter is 1 and 3 units away from
197 and
199, which are the smallest k such that |M(k)| = n, for n of 7 and 8 (respectively),[183] where the sum between the fifteenth pair of
twin primes[8]197 + 199 = 396 is the third
digit-reassembly number in decimal[127] equal to the sum of the first two (132, 264).
^Another relevant example is the thirtieth prime number 113,[9] where thrice its value is
339, whose average of divisors is 114.[10][11] The three largest of a total four numbers that have a sum-of-divisors of 456 (the only number to have an aliquot sum of 744)[14] have arithmetic means that are 114 (they are 222, 302, 339, and 407),[10][11] with 339 between adjacent terms that sum to the 127th prime number and 31st
super-prime,
709.[61] In the list of numbers to reach record amplitude values between zeroes in the Mertens function, the first few terms are (1, 31, 114, 199, 443, 665, 1109, ...);[187] the fourth term is the 46th prime number (199),[9] an index that is itself the 31st in the list of composites,[28] with 443 having an arithmetic mean of divisors of 222 (since it is prime).[10][11] After 1109, the next term is 1637, which is 66516 in
hexadecimal; 31 is the sum-of-divisors of
16, the first of only two numbers (the other being 25),[32] where 31 is also the sum of prime factors of 665 = 5 × 7 × 19.[188] Otherwise, the difference 709 − 339 = 370 represents the second-largest non trivial
Armstrong number equal to the sum of the
cubes of their digits (of only four in
decimal, aside from 0 and 1),[189][190] which follows
153, where
407 is the largest; else, 709 and 339 have a range of 371 integers, which is the third-largest Armstrong number equal to the cube of its digits. The 153rd non-trivial and unitary
arithmetic number after one is 222, where (153, 154) form the sixth
Ruth-Aaron pair[191] with a common sum of distinct prime factors of 20[192] (the eleventh composite, where the first Ruth-Aaron pair is between 5 and 6). The sum of the composite index of 20, alongside the twentieth composite and prime numbers, is in equivalence with 11 + 32 + 71 = 114. Where thrice 20 is 60, half 114 is 57, which represents the composite index of
80, with 57 the 40th composite number.[28] Ten is the sum between sums of distinct prime divisors (of 5) of the first two Ruth-Aaron pairs, where the second pair is (24, 25), followed by (49, 50).[191]
^742 is the seventhicosahedral number, following 456.[148] The sum of all non-unitary icosahedral numbers up through 742 is in equivalence with 12 + 48 + 124 + 255 + 456 + 742 = 1637, which is the eighth number to reach a record amplitude between zeroes in the Mertens function (of −
16).[73][183] 742 is also the thirty-eighth
central polygonal number,[36] and the number of partitions of 30 into divisors of 30.[193] On the other hand, 742 is the eighth
Keith number following 197,[194] which is the smallest number to reach an amplitude of
7 for M(n).[183]
The number of complete non-ambiguous trees of size 5.[209] These trees are defined as labelled binary trees , where each left (right, child) label appears once, and where a left (resp. right) label ancestor to is labeled strictly greater than .[210][211]
The number of connected regular graphs with 18 nodes and girth at least 5.[212]
The number of
closed walks at a vertex of the complete graph on nine nodes .[213]
Otherwise, 456 appears in the toothpick sequence at the twenty-second generation.[218]Where 114 is the average of divisors of three of four numbers (302, 339, and 407) to have a sum-of-divisors of 456, 114 is also the
polynomial degree of seven-cycle
logistic mapr7.[219][139]
^264 is the arithmetic mean of the sixteen divisors that belong to the fourth
primary pseudoperfect number1806 (largest of six numbers that have this value),[220][10][11] that is also the product of the first four terms (2, 3, 7, 43) in
Sylvester's sequence of numbers whose reciprocals most rapidly converge to
1 (only the first four primary pseudoperfect numbers up to 1806 are products of consecutive terms in Sylvester's sequence). The aliquot sum of 1806, on the other hand, is 2418,[14] which is the seventh number after 744 where σ(φ(n)) is equal to n;[131] also, the sum-of-divisors of 264 is 720, where the sum-of-divisors of 720 is 2418.[32] The third-largest primary pseudoperfect number is
42,[220] the second-smallest sphenic number after 30,[221] whose aliquot sum it equals; 42 is specifically the
reduced totient value of 1806 (that of 744 is 30, on the other hand),[26] and the composite index of 60, the smallest number with twelve divisors (12 is also the arithmetic mean of divisors of 42, and its totient value).[10][11][5] The fifth
Mersenne prime after 3, 7, 31 and 127 is
8191, which is the 1028th prime number, an index value whose sum-of-divisors is 1806, with a totient of 512 = 83[5] and a reduced totient of 256 = 28.[26]372 (half of 744) is the number of partitions of 42 into prime parts, where 124 is the number of such partitions of
32, and 52 (the prime index of 239, the number of totatives of 744 that are either prime or composite)[5] of 25.[17] Also, 372 is the thirtieth
untouchable number (followed by 406, the index of 744 as a pernicious number), where 248 is the sixteenth (and 124 the seventh), alongside 52 the third.[150] 1806 is also the only
Bernoulli numberBn such that its
denominator is also n,[222] where the first four denominators in fractions other than 1/1 that have numerators of 1 are 2, 6, 30, 42, 30 (five numbers, with 5/66 the following Bernuolli number,[223] where 66 is the third
sphenic number); in this sequence 6 is also the first
perfect number[81] and
unitary perfect number (like 60, which is twice 30),[224] where 2 represents the first primary pseudoperfect number, before 6.
^71 is the twentieth prime number and 31 the eleventh. In turn,
20 is the eleventh composite number[28] that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163. 71 is also part of the largest pair of
Brown numbers(71, 7), of only three such pairs; where in its case 72 − 1 = 5040.[248][249] Consequently, both 5040 and 5041 can be represented as sums of non-consecutive
factorials, following 746, 745, and 744;[2] where 5040 + 5041 = 10081 holds an aliquot sum of
611, which is the composite index of 744.[28] 5040 is the nineteenth
superabundant number[250] that is also the largest factorial that is a
highly composite number,[251] and the largest of twenty-seven numbers n for which the
inequalityσ(n) ≥ eγnloglogn holds, where γ is the
Euler–Mascheroni constant; this inequality is shown to fail for all larger numbers if and only if the
Riemann hypothesis is true (known as Robin's theorem).[242] 5040 generates a sum-of-divisors 19344 = 13 × 31 × 48 that itself contains four divisors in proportion with 744 (and therefore, divisors also in proportion with 248 as well); which makes it one of only two numbers out of these twenty-seven integers n in Robin's theorem to hold σ(n) such that 744m | σ(n) for any subset of divisors m of n; the only other such number is 240:[252]
19344 ÷ 26 = 744
9672 ÷ 13 = 744
1488 ÷ 2 = 744
Where also 19344 ÷ 78 = 248, with 248 and 744 respectively as the 24th and 30th largest divisors (where in between these is 403 = 13 × 31, which is the middle indexed composite number congruent ±1 mod 6 less than 744 that is not part of its composite totatives); 2418, the 35th largest, is the seventh number n after 744 such that σ(φ(n)) is n.[131] Furthermore, in this sequence of integers in Robin's theorem, between 240 and 5040 lie four numbers, where the sum between the first three of these 360 + 720 + 840 = 1920 is in equivalence with σ(744). The first number to be divisible by all positive non-zero integers less than 11 is the penultimate number in this sequence
2520, where 2520 − 840 − 720 = 960 represents a Zumkeller half from the set of divisors of 744,[80] with σ(720) = 2418[14] (and while 720 + 24 = 744 = 6! + 24, where 720 is the smallest number with thirty divisors,[12] equal to 1176 − 456, a difference between the aliquot sum of 744, and the only number to have an aliquot sum of 744; wherein 720 is also in equivalence with σ(264) = 456 + 264).[14][32]5040 = 7! = 10 × 9 × 8 × 7 is divisible by the first twelve non-zero integers, except for 11.
^163 is also the sum between prime indexes (81,
82) that belong to numbers adjacent to the twenty-second
interprime420 = 101 + 103 + 107 + 109, the twenty-sixth number that is the sum of four consecutive primes[253] that is also the 45th number to return 0 for the Mertens function.[92] It is the sixth such interprime to yield prime values from sums of prime indices of adjacent numbers, after (138, 72, 12, 6, 4) and preceding 432, the abundancy of 744.[24] 420, with a
Euler totient of 96,[5] is half of
840, the smallest number with exactly thirty-two divisors,[12] and twice
210, the fourth
primorial(2 × 3 × 5 × 7), where the previous three primorials sum to 2 + 6 + 30 =
38,[254] in equivalence with the prime index of 163.[9] Furthermore, the 163rd composite number is 210,[28] where 210 is thrice
70. Seventy is the fifth
pentatope number,[255] that represents the sum of prime factors (inclusive of multiplicities) in what is described as the least-interrelated sporadic group, Janko group J3, vis-à-vis all other sporadic groups (including the pariah groups), and their relevant algebraic structures. 70 is primarily used in the construction of the
Leech lattice in twenty-four dimensions, that involves the only non-trivial solution (i.e., 1) to the
cannonball problem, equal to the sum of the squares of the first twenty-four integers; 70 is also equal to the number of
conformal field theories in the form of
vertex operator algebras with
central charge of 24 with weight 1 (and aside from V0, see
§ E8 and the Leech lattice), that are rooted in the Leech lattice, of which the largest two exist in dimension seven hundred and forty-four. 210 is, furthermore, the first non-trivial 71-
gonal number of the form P(s,n) = (s − 2)n2 − (s − 4)n/2 (for the nth polygonal number with s sides), that precedes
418, the sum of integers between 13 and 31, inclusive. Importantly, 210 is the largest number n where the number of distinct representations of n as the
sum of two primes is at most the number of primes in the interval (n/2, n – 2).[256]420, which is the eleventh self-convolution of Fibonacci numbers,[226] preceding 744, is the totient value of 473, the first non-trivial number to hold this value (after 421). Twice 744 is 1488, which lies between the 50th pair of twin primes (1487, 1489),[8] whose successive prime indices (236, 237) add to 473. 1488 is the 235th average of successive odd primes,[59] where 235 is the sixth self-convolution of Fibonacci numbers, preceding 420,[226] with 298 the 235th composite, where 372 (half 744) is the 298th (235 is also the 183rd composite).[28] The fiftieth composite number, on the other hand, is 70.[28]
^Worth noting, 126 is the sum of prime indices of the first class of three-digit
permutable primes in decimal (
113, 131, 311),[264] respectively the thirtieth, thirty-second, and sixty-fourth prime numbers[9] (where
64 is twice
32, and the 45th composite, with
45 itself the 30th composite);[28] and this sum of prime numbers is equivalent with 555. Furthermore, 126 is the seventh successive composite index starting with 13 (i.e., 13, 22, 34, 50, 70, 95, 126).[265] 126 is equal to the sum of the composite index of
22 (
13), and the twenty-second composite and prime numbers (
34,
79), whose sum generated between the latter two is 113. 13 and 79 are also first members in their respective paired permutable primes in base ten (13, 31) and (79, 97), respectively the smallest and largest two-digit pairs, where 11 is the only two-digit permutable prime without a distinct dual pair with differing permuting digits.[7] Where 555 is generated from the first class of three-digit permutable primes in decimal, and prime and composite indices associated with 22, 131 and 126 sum to 257, which is the 55th indexed prime[9] and tenth prime number that is not a cluster prime[89] (and where, 13110 has a
digit sum of 5). Also, 126 + 125 =
351, a value in equivalence with the twenty-sixth triangular number,[6] with 125 the
cube of 5.
^Regarding 992, the numbers 336 (the totient of 1176,[5] which is the sum-of-divisors of 744), 496 (the 31st triangular number and third perfect number),[6][81] and 525 — the sum of all prime numbers that divide the orders of the twenty-six (or twenty-seven) sporadic groups,[272] equal to the sum of the dimensions of all five
exceptional Lie algebras[273] — are the first three of only four composites to have a sum-of-divisors of 992.[32] Furthermore, the fifth such number is 775, whose composite index is 637, equal to the sum of all prime factors (inclusive of multiplicities) in the
order of the largest sporadic group, |1| = 246 × 320 × 59 × 76 × 112 × 133 × 17 × 19× 23 × 29 × 31 × 41 × 47 × 59 × 71.[235]: pp.244–246 In Chowla's function (that counts the sum-of-divisors of n except for 1 and and n), 525 holds a value of 466[274] that is the prime index of 3313,[9] equal to the sum of all seven numbers (240, 350, 366, 368, 575, 671, and 743) to hold a sum-of-divisors of 744;[32] with 240 the totient of 525 as well (the 13th of 31 numbers to hold this value).[5]
^More deeply, for the theta series of 744 is a prime-indexed coefficient over its first six indices less than 104 (respectively, the 458th, 526th, 742nd, 799th, 842nd, and 1141st prime indices, with a sum of 4058; the latter in equivalence with 1141 =
7 ×
163).[282] The first composite coeff. index in the series with coefficient 744 is its seventh index 9251 = 11 × 292, whose sum of prime factors, inclusive of 1, is
70 (which is of algebraic significance in terms of the construction of the twenty-four-dimensional
Leech lattice). 456, and 1176, the
aliquot sum of 744,[14] are also prime-indexed coefficients over their first two coefficient indices (346th, 364th, and 1098th, 1159th, respectively).[282]
^Where the smallest non-unitary
pentagonal pyramidal number is
6, the eleventh is 726 = 6! + 6, and the twenty-fourth is
7200,[308] which is a number with a Euler totient value of 1920,[5] and a reduced totient of 120.[26]
^Such that it can be expressed using just its digits (which are only used once, and from left to right) alongside the
operators+, -, ×, ÷, ab, √, ! (with concatenation allowed). This, in likeness of 144 = (1 + 4)! + 4!, that is the
Euler totient of 456.[5] Where the totient of 744 is 240, that of 456 is 144.[5]187 is the composite index of 240, where 187 is the 144th composite number.[28] In turn, the sum-of-divisors of 187 is 216 = 63,[32] which is the 168th composite number. Also, the reduced totient of 456 is 36.[26]
^The magic square is:
This is the second-smallest magic constant for a 6 × 6 magic square consisting of thirty-six consecutive prime numbers, where the sum between the smallest and largest prime numbers in this square is equal to 41 + 223 = 264. The smallest such constant is 484 = 222[312] whose aliquot sum of 447 is the reverse permutation of the digits of 744 in decimal;[14] specifically,
22 and 264 are respectively the twelfth and fourteenth numbers n whose squares are
undulating in decimal, while the thirteenth and penultimate such known number is 26.[313][314][315]: pp.159, 160 The smallest possible magic constant of an n × n magic square consisting only of distinct prime numbers is 120, from an n of 4,[316][317] a value equal to the arithmetic mean of all sixteen divisors of 744;[10][11] otherwise, for n = 6, the smallest magic constant for a six-by-six square with distinct prime numbers is 432,[316] also the abundance of 744.[24] An 11 × 11 magic square that is normal has a magic constant of
671,[318] which is the sixth and largest composite number to have a sum-of-divisors equal to 744.[32]
^Otherwise, 992 = 248 + 744 is the least possible number of
diagonals of a
simple convex
polyhedron with thirty-six
faces; for sixteen and twenty faces, there are respectively a minimum of 132 and 240 diagonals[320] (values which represent the prime index of the largest prime
totative of 744 and the count of all its totatives),[5] where 132 + 240 = 372, or half 744 (alongside respective indices that generate a sum of 36).
^On the other hand, 456 (the only number to have an aliquot sum of 744)[14] is the maximum number of unit squares — only joined at corners — that can be inscribed inside a square (of area 1600u2).[322] Adding 1 to the value of this area, 1601 = 1600 + 1, is the largest of forty
lucky numbers of Euler, that is also the difference between 1804 − 203, values that represent the first and third longest sides of primitive
Heronian tetrahedra with four congruent
triangle faces, with the second such number equaling 144 + 744 = 888.[323][324]
^The j–invariant can be computed using
Eisenstein seriesE4 and E6, such that: j(𝜏) = 1728 E4(𝜏)3/E4(𝜏)3 − E6(𝜏)2 , where E4(𝜏) = 1 + 240 Σ∞ n = 1(n3qn/1 − qn) and E6(𝜏) = 1 − 504 Σ∞ n = 1(n5qn/1 − qn), with q = exp(2πi𝜏). The respective q–expansions of these two Eisenstein series have coefficients whose numerical values are in proportion to 240 and −
504, respectively;[228][229] where specifically the sum and difference between the absolute values of these two numbers is 240 + 504 = 744 and 504 − 240 = 264. Furthermore, when considering the only smaller even (here, non-modular) series E2(𝜏) = 1 − 24 Σ∞ n = 1(nqn/1 − qn), the sum between the absolute value of its constant multiplicative term (24) and that of E4(𝜏) (240) is equal to 264 as well. The 16th coefficient in the expansion of E2(𝜏) is −744, as is its 25th coefficient.[230] Alternatively, the j–invariant can be computed using a
sextic polynomial as: j(λ) = 28 × (λ2 − λ + 1)3/λ2(λ − 2)2 where λ represents the λ−
modular function, with 256 = 28.[231]
^The smallest sporadic group is
Mathieu groupM11, with an irreducible complex representation in ten dimensions,[237] that is one of five first generation sporadic groups.[238] Its
group order is 7920 = 8 × 9 × 10 × 11 (one less than the 103 indexed prime number),[235]: pp.244–246 [9] with a Euler totient of 1920 = σ(744).[5]M11 also holds a lowest
five-dimensional faithful representation over the field with three elements, the lowest of any such group.[237] On the other hand, M12 is the second-largest Mathieu group and sporadic group by order,[235]: pp.244–246 and more specifically the thirty-first largest non-
cyclic group (where M11 is the fifteenth largest),[239][240] with group order 26 × 33 × 5 × 11 = 95040 whose Euler totient (23040)[5] is divisible by 1920 twelve-times over (as its 50th largest divisor greater than 1). The nineteenth and twentieth largest non-cyclic groups are A3(2) ≃ A8 and A2(4) of orders 20160 = 26 × 32 × 5 × 7, the latter with larger
outer automorphism groupD12; 20160 is the twenty-third
highly composite number,[241] that is divisible by both 960 (the Zumkeller half of 744),[80] and 5040 (the largest of twenty-seven numbers in Robin's theorem to hold
inequalityσ(n) ≥ eγnloglogn, see latter points discussing this sequence).[242] It is one less than the largest number (20161, the 1456th indexed number) which cannot be represented as the sum of two
abundant numbers.[243] Otherwise, the
Tits groupT, the only finite simple group that can classify as of
Lie type or sporadic, fits as the fifth-largest sporadic group, whose group order 211 × 33 × 52 × 13 = 17971200 holds a totient of 4423680, divisible by the sum-of-divisors of seven-hundred and forty-four 2304 times (where its 62nd largest divisor is 1728, followed by 1920). Of all (here, twenty-seven) sporadic groups, only the seventh largest
Janko groupJ3 has an order, |J3| = 27 × 35 × 5 × 17 × 19 = 50232960,[235]: pp.244–246 whose totient value is not divible by 1920: 11943936 ÷ 1920 = 6022.8; where specifically its group order has a sum of prime factors (inclusive of multiplicities) equal to 70. Where M11 and M12 are respectively the fifteenth and thirty-first largest indexed non-cyclic groups by group order, these indices sum to 15 + 31 = 46, where the third-largest Mathieu group and fourth-largest sporadic group M22 is the forty-sixth such largest non-cyclic group.[239][240]46 is the largest even number that cannot be represented as the sum of two abundant numbers,[243] that is also equal to the total number of
maximal subgroups of the
Friendly Giant,[244] which collectively holds 20 of 26 sporadic groups as
subquotients (strictly).[235]: pp.244–246 46 is the thirty-first composite,[28] whose aliquot sum is 26, the only number to hold this value.[14]
^The sequence of self-convoluted Fibonacci numbers starts {0, 0, 1, 2, 5, 10, 20, 38, 71, 130, 235, 420, 744, 1308, 2285, 3970, 6865, 11822...}.[226] The sum of the first seven terms (from the zeroth through the sixth term) is equal to 38, which is equivalent to the a(n) = 7 member in this sequence. Taking the sum of the three terms that lie between 71[xxx] and 744 (i.e. 130, 235, 420) yields 785 =
282 + 1, whose
aliquot sum is 163,[14][xxxi] the thirty-eighth prime number.[9]785 is the 60th number to return 0 for the
Mertens function, which also includes 163, the 13th such number.[92] 785 is also the number of
irreducibleplanted trees (of root vertex having degree one) with six
leaves of two
colors.[257]1106, the first member in the sum of consecutive composite numbers (in equivalence with 2229 = 743 × 3) that are divisible by the twenty-fifth prime number 97,[258] is the smallest number to return a value of 14 in the inverse
Mertens function,[183] where 1106 lies between the 81st pair of
sexy primes(1103, 1109).[259]14 is the
floor (and
nearest integer)[260] of the
imaginary part of the first non-trivial zero in the
Riemann zeta functionζ, and the fourteenth indexed floor value is 60[261][262] — sixty is the smallest number with exactly twelve divisors, and there are only two numbers which have a
sum-of-divisors of 60:
24 and
38,[32] whose sum is 62 (which is the tenth largest divisor of 744); also, three of nine numbers with totients of 60 are also divisors of 744 (93, 124, and 186, that add to 403).[5] On the other hand, the difference 38 − 24 = 14, where the sixtieth floor value in the imaginary part of its nontrivial zero in ζ is the largest Heegner number163.
^163 is the thirty-eighth indexed prime and
67 the nineteenth (with the
biprime38 = 2 ×
19), for values d. In the approximation for the almost integer containing the fourteenth prime number as the seventh Heegner number
43 for d, 960 is the smallest number with exactly twenty-eight divisors,[12] equivalent to the sum of either of two sets of divisors of 744 that collectively add to 1920, as mentioned in
§ Totients. Likewise, in the approximation for the almost integer containing the eighth Heegner number 67 for d,
5280 is equal to the sum between 240 and 5040, which are the only two numbers in the set of twenty-seven integers in Robin's theorem for the Riemann hypothesis that have a set of divisors whose sums are divisible by 744;[252] 5280 also lies between the 131st pair of twin primes (5281, 5279),[8] respectively the 701st and 700th prime numbers, where 5281 is the 126th super-prime.[61][xxxii]
^In the list of thirteen integers n ≤ 1000 that yield almost integers — with
nearness|nint(x) − x| ≤ 10−3 for values x ≡ eπ√n — the smallest n of these is 25, that is also the smallest composite totative of 744. Less than 103, the largest such n is 719 = 6! − 1, where
719 represents the 128th indexed prime number;[9] the sum generated by these two n is in equivalence with 719 + 25 = 744. The middle indexed value in between these two bounds is 148, the twelfth square-free positive integer d over the negated imaginary quadratic field of class number 2, followed by 163 and
232, the latter of which is the fourteenth square-free positive integer d over the imaginary quadratic field √−d of class number 2 (only these two numbers 148 and 232 for d in this field of class number 2 yield almost integers with |nint(x) − x| ≤ 10−3, where 163 is the largest over the same field with class number 1).[267] In this same list, the three distinct non-supersingular primes (37, 43, 67) that only divide orders of pariah groups are also n (less than 148) that yield almost integers of the form eπ√n. In the almost integer representation for eπ√148 specified, 8 × 103 = j(√(2i)) = 203, the class 1j−invariant of order d = −8,[268] that is also a root in Ψ2(X, X) = −(X − 8000) × (X + 3375)2 × (X − 1728), a polynomial for supersingular 2-isogeny graphs with
loops (it is also a root in Ψ3).[269][270] The remaining two n (58, 74) in the list of these almost integers with nearness |nint(x) − x| ≤ 10−3 of n less than 148 generate a sum of 58 + 74 = 132 equal to the prime index of 743,[9] the largest prime totative of 744 (and, where 2 × 37 = 74 = 148 ÷ 2).
^For almost integers with nearness |nint(x) − x| ≤ 10−2, the smallest such number is eπ√6 with n of 6, approximately equal to 2197.99087.[271] Whereas 2198 holds eight divisors that produce an arithmetic mean of 474[10][11] (and where 474 holds a sum-of-divisors equal to 960,[32] also the Zumkeller half from the set of divisors of 744),[80] 2199 is the sixteenthperfect totient number,[34] with an aliquot sum of
737[14] equivalent with the largest composite totative of 744. Also, 2197 = 133. There are a total of twenty-six almost integers with |nint(x) − x| ≤ 10−2 where x ≡ eπ√n and n ≤ 103, where the largest n holds a value of
986.[271] The sum between the upper and lower bounds n less than one thousand with this almost integer degree of nearness is equal to 6 + 986 = 992 = 744 + 248.[xxxiii] Otherwise, the difference between the largest class 1 and 2 square-free integers is 427 − 163 = 264. Regarding square-free integers of class number h = 3, there are a total of sixteen (or twenty-five when including non-maximal orders),[275] the largest with value of 907 that is the 155th indexed prime number,[9] equal to the sum 163 + 744, that is also the greater of consecutive primes (887, 907) that generate the seventh largest record prime gap (of 20).[75][76][77][78][79] The previous greatest prime gap is of 18 set by (523,
541) (following the thirtieth and thirty-first prime numbers 113 and 127 with a gap of 14), the 99th and 100th prime numbers,[9] the former also the composie index of 132,[28] and the latter the tenth
star number[146] and 53rd number to return 0 for the
Mertens function (where 427 is the 50th, and 163 the 13th).[92] Of the square-free integers of class number three, only
59 produces an almost integer of the form x ≡ eπ√n with |nint(x) − x| ≤ 10−2 (for integers of class number h = 4, the largest such number is 1555 — whose
square divides 61554[276] — while the ninth such largest discriminant is 155).[277] In the full list of largest square-free positive integers with class numbers h ≤ 100,[278][279] ten of these maximum values are held by larger integers when non-maximal orders in their respective classes are included, where nine of these are uniquely biprimes divisible by 163 (the only exception is the largest value for square-free integers of class number 16); the largest of these in this bound is 821683 = 712 × 163 of class number h = 70.[263]: 18−20
^+ 4≅ as a lattice is the union of two lattices,[281] where the associated theta series of has 744 as its 50th indexed coefficient (as with theta series of ),[284][280] and where its twenty-fifth coefficient is
248, which is the most important divisor of 744. Also, in this theta series of , the preceding 49th coefficient is
456, the only number to hold an
aliquot sum of 744,[14] where 744 − 456 =
288, a value that is the number of
cells in the
disphenoidal 288-cell, whose 48
vertices collectively represent the twenty-four
Hurwitz unit quaternions with
norm squared 1, united with the twenty-four vertices of the dual
24-cell with norm squared 2. For the theta series of , that is realized in the
16-cell honeycomb, all 25 × 2m indexed coefficients (i.e. 25, 50, 100, 200, 400, ...) are 744.[284] For both the theta series of and , the 456th coefficient is
1920, the
sum-of-divisors of 744.[32] Also, for the theta series of the four-dimensional lattice, coefficient index 288 is the 97th non-zero norm, coeff. index 456 the 153rd (an index that represents the seventeenth
triangular number), and coeff. index 744 the 250th;[283] the latter, a coefficient index that is the largest of only four numbers to hold a
Euler totient of
100: 101, 125, 202, and
250[5] — the smallest of these is the twenty-sixth prime number, while the largest 250 = 2 × 53 has a sum of prime factors that is
17, the seventh prime number;[9] and where 202 = 49 + 153 and 250 = 153 + 97, with 549 = 49 + 97 + 153 + 250, the 447th indexed composite number.[28][xxxiv]
^The
Petrie polygon of the latter two which are rooted in
icosahedral symmetry is the thirty-sided
triacontagon, while the former has a
dodecagon as its like-polygon, whose
central angle is of thirty degrees (and internal angle of 150 degrees). The twenty-fourth prime is
89,[9] which is the index of 120 in the list of composites.[28] In a 600-cell fit precisely seventy-five 16-cells and seventy-five
tesseracts, where there also fit twenty-five
24-cells. Where twenty-five 24-cells also fit inside a
24-cell honeycomb as a unit (with twenty-four 24-cells surrounding a single 24-cell), whose symmetry is rooted in , a different arrangement of twenty-five 24-cells fits in a 600-cell with symmetry. , in eight-dimensional space, has a
Coxeter number of 30, and a symmetry that traces back to the 600-cell via the
icosians.
^The
8 lattice holds two hundred and forty
root vectors that are represented by the
vertex arrangement of the
421 polytope, whose
Petrie polygon is a thirty-sided
triacontagon,[293][294] where 30 is the
Coxeter numberh of
Coxeter groupE8. Also, Coxeter numbers of simple reflexions in E6 and E7 are of orders 12 and 18, together equivalent to the Coxeter number of E8;[285]: 234 where E6 and E7embed inside E8. Four-dimensional H4hexadecachoric symmetry also contains a Coxeter number of
30, where H4 is the higher-dimensional analogue of icosahedral symmetry Ih. This 8 lattice structure with 240 root vectors can be constructed with 120 quaternionic
unit icosians that form the vertices of the four-dimensional
600-cell, whose symmetries are rooted in three-dimensional
icosahedral symmetryIh of order 120,[295] a value equal to the total number of reflections of
Coxeter groupE8.[285]: 226–232 In total, a
regular icosahedron and
dodecahedron contain thirty-one axes of symmetry; six five-fold, ten three-fold, and fifteen two-fold.[296] Where the arithmetic mean of divisors of 744 is 120,[10][11] with a largest prime factor of 31 and a reduced totient λ(n) of 30,[26] the number of integers that are
relatively prime with and up to 744 is 240,[5] with the sum-of-divisors σ(240) = 744.[287]: p.21
^VE3 8 is isomorphic to the
tensor productVE8⊗ VE8⊗ VE8; also, affine structure D16 is different from D+ 16, which is associated with even positive definite
unimodular lattice+ 16.[300][301] E3 8,1 and D16,1E8,1 are associated with codes e3 8 and d16e8 that are two of only nine in-equivalent doubly even self-dual codes of length 24 and weight 4.[302][303] The largest of these VOAs VD24 is realized in dim 1128, where successively halving its dimensional space leads to a 70+1⁄2–dimensional space.
"Of particular curiosity is the less well-known fact – in parallel to the above identity – that the constant term of the j-invariant, viz., 744, satisfies 744 = 3 × 248. The number 248 is, of course, the dimension of the adjoint of the largest exceptional algebra [𝖊8]. In fact, that j should encode the presentations of [𝖊8] was settled long before the final proof of the Moonshine conjectures. This relationship between the largest sporadic group and the largest exceptional algebra would connect the McKay Correspondence to Moonshine and thereby weave another beautiful thread into the panoramic tapestry of mathematics.": p.4
"16 Incidentally, the reader is also alerted to the curiosity that σ1(240) = 744.": p.21 (note)
"In particular, 4124 = 3875 + 248 + 1 and 34752 = 30380 + 3875 + 2 · 248 + 1, where 248, 3875 and 30380 are all dimensions of irreducible representations of E8().": p.6