126 (number)

From HandWiki
Short description: Natural number
← 125 126 127 →
Cardinalone hundred twenty-six
(one hundred twenty-sixth)
Factorization2 × 32 × 7
Divisors1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126
Greek numeralΡΚϚ´
Roman numeralCXXVI
Base 363I36

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

In mathematics

As the binomial coefficient [math]\displaystyle{ \tbinom{9}{4} }[/math], 126 is a central binomial coefficient[1] and a pentatope number.[2] It is also a decagonal number[3] and a pentagonal pyramidal number.[4] As 125 + 1 it is σ3(5), the fifth value of the sum of cubed divisors function,[5] and is a sum of two cubes.[6]

There are exactly 126 crossing points among the diagonals of a regular nonagon,[7] 126 binary strings of length seven that are not repetitions of a shorter string,[8] 126 different semigroups on four elements (up to isomorphism and reversal),[9] and 126 different ways to partition a decagon into even polygons by diagonals.[10] There are exactly 126 positive integers that are not solutions of the equation

[math]\displaystyle{ x=abc+abd+acd+bcd, }[/math]

where a, b, c, and d must themselves all be positive integers.[11]

It is the fifth 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.[12]

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


  1. Sloane, N. J. A., ed. "Sequence A001405 (Central binomial coefficients)". OEIS Foundation. https://oeis.org/A001405.  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))". OEIS Foundation. https://oeis.org/A000332. 
  3. (Deza Deza), pp. 2–3 and 6; Sloane, N. J. A., ed. "Sequence A001107 (10-gonal (or decagonal) numbers)". OEIS Foundation. https://oeis.org/A001107. 
  4. (Deza Deza), pp. 93, 211. Sloane, N. J. A., ed. "Sequence A002411 (Pentagonal pyramidal numbers)". OEIS Foundation. https://oeis.org/A002411. 
  5. Sloane, N. J. A., ed. "Sequence A001158 (sigma_3(n): sum of cubes of divisors of n)". OEIS Foundation. https://oeis.org/A001158. 
  6. Sloane, N. J. A., ed. "Sequence A003325 (Numbers that are the sum of 2 positive cubes)". OEIS Foundation. https://oeis.org/A003325. 
  7. Sloane, N. J. A., ed. "Sequence A006561 (Number of intersections of diagonals in the interior of regular n-gon)". OEIS Foundation. https://oeis.org/A006561. 
  8. Sloane, N. J. A., ed. "Sequence A027375 (Number of aperiodic binary strings of length n; also number of binary sequences with primitive period n)". OEIS Foundation. https://oeis.org/A027375. 
  9. 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))". OEIS Foundation. https://oeis.org/A001423. 
  10. Sloane, N. J. A., ed. "Sequence A003168 (Number of blobs with 2n+1 edges)". OEIS Foundation. https://oeis.org/A003168. 
  11. 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)". OEIS Foundation. https://oeis.org/A027566. . See OEIS:A027563 for the list of these 126 numbers.
  12. J. D. Koninck, Those Fascinating Numbers, transl. author. American Mathematical Society (2008) p. 40.
  13. Emsley, John (2011), Nature's Building Blocks: An A-Z Guide to the Elements, Oxford University Press, p. 592, ISBN 9780199605637, https://books.google.com/books?id=4BAg769RfKoC&pg=PA592