Natural number
Cardinal one hundred twenty-six
Ordinal 126th (one hundred twenty-sixth)
Factorization 2 × 32 × 7
Divisors 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Greek numeral ΡΚϚ´
Roman numeral CXXVI
Binary 11111102
Ternary 112003
Senary 3306
Octal 1768
Duodecimal A612
Hexadecimal 7E16
126 (one hundred [and] twenty-six ) is the
natural number following
125 and preceding
127 .
In mathematics
As the
binomial coefficient
(
9
4
)
{\displaystyle {\tbinom {9}{4}}}
, 126 is a
central binomial coefficient , and in
Pascal's Triangle , it is a
pentatope number .
[1]
[2] 126 is a
sum of two cubes , and since 125 + 1 is σ3 (5), 126 is the fifth value of the
sum of cubed divisors function .
[3]
[4]
126 is the fifth
S
{\displaystyle {\mathcal {S}}}
-perfect
Granville number , and the third such not to be a perfect number. Also, it is known to be the smallest Granville number with three distinct prime factors, and perhaps the only such Granville number.
[5]
126 is a
pentagonal pyramidal number and a
decagonal number .
[6]
[7] 126 is also the different number of ways to partition a
decagon into even polygons by
diagonals , and the number of crossing points among the diagonals of a regular
nonagon .
[8]
[9]
There are exactly 126
binary strings of length seven that are not repetitions of a shorter string, and 126 different
semigroups on four elements (up to
isomorphism and reversal).
[10]
[11]
There are exactly 126 positive integers that are not solutions of the equation
x
=
a
b
c
+
a
b
d
+
a
c
d
+
b
c
d
,
{\displaystyle x=abc+abd+acd+bcd,}
where a , b , c , and d must themselves all be positive integers.
[12]
126 is the number of
root vectors of
simple
Lie group
E7 .
In physics
126 is the seventh
magic number in
nuclear physics . For each of these numbers, 2, 8, 20, 28, 50, 82, and 126, an atomic nucleus with this many protons is or is predicted to be more stable than for other numbers. Thus, although there has been no experimental discovery of element 126, tentatively called
unbihexium , it is predicted to belong to an island of stability that might allow it to exist with a long enough
half life that its existence could be detected.
[13]
See also
References
^
Sloane, N. J. A. (ed.).
"Sequence A001405 (Central binomial coefficients)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. See also
OEIS:A001700 for the odd central binomial coefficients.
^
Deza, Elena ;
Deza, M. (2012), "3.1 Pentatope numbers and their multidimensional analogues", Figurate Numbers , World Scientific, p. 162,
ISBN
9789814355483 ;
Sloane, N. J. A. (ed.).
"Sequence A000332 (Binomial coefficients binomial(n,4))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A003325 (Numbers that are the sum of 2 positive cubes)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A001158 (sigma_3(n): sum of cubes of divisors of n)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
de Koninck, Jean-Marie (2008).
Those Fascinating Numbers . Translated by de Koninck, J. M. Providence, RI:
American Mathematical Society . p. 40.
ISBN
978-0-8218-4807-4 .
MR
2532459 .
OCLC
317778112 .
^
Deza & Deza (2012) , pp. 93, 211.
Sloane, N. J. A. (ed.).
"Sequence A002411 (Pentagonal pyramidal numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Deza & Deza (2012) , pp. 2–3 and 6;
Sloane, N. J. A. (ed.).
"Sequence A001107 (10-gonal (or decagonal) numbers)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A003168 (Number of blobs with 2n+1 edges)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A006561 (Number of intersections of diagonals in the interior of regular n-gon)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A027375 (Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A001423 (Number of semigroups of order n, considered to be equivalent when they are isomorphic or anti-isomorphic (by reversal of the operator))" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation.
^
Sloane, N. J. A. (ed.).
"Sequence A027566 (Number of numbers not of form k_1 k_2 .. k_n (1/k_1 + .. + 1/k_n), k_i >= 1)" . The
On-Line Encyclopedia of Integer Sequences . OEIS Foundation. . See
OEIS:A027563 for the list of these 126 numbers.
^ Emsley, John (2011),
Nature's Building Blocks: An A-Z Guide to the Elements , Oxford University Press, p. 592,
ISBN
9780199605637
100,000
1,000,000
10,000,000
100,000,000
1,000,000,000