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126 (one hundred twenty-six) is the natural number following 125 and preceding 127.

In mathematics

As the binomial coefficient ${\displaystyle {\tbinom {9}{4}}}$, 126 is a central binomial coefficient, and in Pascal's Triangle, it is a pentatope number. 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.

126 is the fifth ${\displaystyle {\mathcal {S}}}$-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.

126 is a pentagonal pyramidal number and a decagonal number. 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.

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).

There are exactly 126 positive integers that are not solutions of the equation

${\displaystyle x=abc+abd+acd+bcd,}$

where a, b, c, and d must themselves all be positive integers.

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.

References

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.
6. 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.
7. 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.
8. Sloane, N. J. A. (ed.). "Sequence A003168 (Number of blobs with 2n+1 edges)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. 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.
10. 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.
11. 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.
12. 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.
13. Emsley, John (2011), Nature's Building Blocks: An A-Z Guide to the Elements, Oxford University Press, p. 592, ISBN 9780199605637

• Integers