Natural number
138 (one hundred [and] thirty-eight ) is the
natural number following
137 and preceding
139 .
In mathematics
138 is a
sphenic number ,
[1] and the smallest product of three primes such that in
base 10 , the third prime is a
concatenation of the other two:
2
⋅
3
⋅
23
{\displaystyle 2\cdot 3\cdot 23}
.
[a] It is also a one-step
palindrome in decimal (138 + 831 = 969).
Four concentric
magic circles , with a magic constant of 138.
138 has eight total
divisors that generate an
arithmetic mean of
36 ,
[2] which is the eighth
triangular number .
[3] While the sum of the digits of 138 is
12 , the product of its digits is
24 .
[4]
138 is an
Ulam number ,
[5] the thirty-first
abundant number ,
[6] and a primitive (
square-free )
congruent number .
[7] It is the third 47-
gonal number .
[8]
As an
interprime , 138 lies between the eleventh pair of
twin primes (
137 ,
139 ),
[9] respectively the 33rd and 34th prime numbers.
[10]
It is the sum of two consecutive primes (
67 +
71 ),
[11] and the sum of four consecutive primes (
29 +
31 +
37 +
41 ).
[12]
There are a total of 44 numbers that are
relatively prime with 138 (and up to),
[13] while 22 is its
reduced totient .
[14]
138 is the
denominator of the twenty-second
Bernoulli number
B
n
{\displaystyle B_{n}}
(whose respective
numerator , is 854513).
[15]
[16]
A
magic sum of 138 is generated inside four
magic circles that features the first thirty-three non-zero
integers , with a 9 in the
center (first constructed by
Yang Hui ).
[b]
The simplest
Catalan solid , the
triakis tetrahedron , produces 138
stellations (depending on rules chosen),
[c] 44 of which are fully symmetric and 94 of which are
enantiomorphs .
[17]
Using two
radii to divide a
circle according to the
golden ratio yields sectors of approximately 138
degrees (the
golden angle ), and 222 degrees.
In science
In media
See also
Notes
^ The only other number less than
1000 in decimal with this property is
777 =
3 ×
7 ×
37 .
^ This magic sum is generated from points that simultaneously lie on circles and diagonals, and, importantly, without including 9 in this sum (i.e. by bypassing it).
^ Using a different set of rules than
Miller's rules for stellating polyhedra. For example, by Miller's rules, the triakis tetrahedron produces a total of 188 stellations, 136 of which are chiral. Using this same set of (Miller) rules, its dual polyhedron, the
truncated tetrahedron , produces only 9 stellations, without including the truncated tetrahedron.
References
^
Sloane, N. J. A. (ed.).
"Sequence A007304 (Sphenic numbers: products of 3 distinct primes.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A003601 (Numbers n such that the average of the divisors of n is an integer)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A000217 (Triangular numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
"138" . Numbers Aplenty . Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A002858 (Ulam numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A005101 (Abundant numbers (sum of divisors of m exceeds 2m).)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A006991 (Primitive congruent numbers.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A095311 (47-gonal numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-05-27 .
^
Sloane, N. J. A. (ed.).
"Sequence A014574 (Average of twin prime pairs.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A006093 (a(n) equal to prime(n) - 1.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A001097 (Twin primes.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A034963 (Sums of four consecutive primes.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A000010 (Euler totient function phi(n): count numbers less than or equal to n and prime to n.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A002322 (Reduced totient function psi(n))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A027642 (Denominator of Bernoulli number B_n.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Sloane, N. J. A. (ed.).
"Sequence A027641 (Numerator of Bernoulli number B_n.)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2023-07-24 .
^
Wenninger, Magnus J. (1983). "Chapter 3: Stellated forms of convex duals".
Dual Models . Cambridge, UK:
Cambridge University Press . pp. 36–37.
doi :
10.1017/CBO9780511569371 .
ISBN
9780521245241 .
MR
0730208 .
OCLC
8785984 .
^
"Who's Afraid Of 138?!" .
Armada Music . Retrieved 2023-07-25 .
^
"Who's Afraid Of 138?!" .
Beatport . Retrieved 2023-07-25 .
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