A000999 5-adic valuation of binomial(2*n,n): largest k such that 5^k divides binomial(2*n, n).

0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1

Offset

0,14

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, Journal für die reine und angewandte Mathematik, Vol. 44 (1852), pp. 93-146; alternative link.

Dorel Miheţ, Legendre's and Kummer's theorems again, Resonance, Vol. 15, No. 12 (2010), pp. 1111-1121; alternative link.

Armin Straub, Victor H. Moll and Tewodros Amdeberhan, The p-adic valuation of k-central binomial coefficients, Acta Arithmetica, Vol. 140, No. 1 (2009), pp. 31-42.

Wikipedia, Kummer's theorem.

Formula

From Amiram Eldar, Feb 12 2021: (Start)

a(n) = A112765(A000984(n)).

a(n) = (2*A053824(n) - A053824(2*n))/4. (End)

Mathematica

Table[IntegerExponent[Binomial[2*n, n], 5], {n, 0, 100}] (* T. D. Noe, Jun 21 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, valuation(binomial(2*n, n), 5))
(PARI) a(n) = my(v=digits(n, 5), c=0); sum(i=0, #v-1, c=(c+v[#v-i]>=3)); \\ Kevin Ryde, Mar 07 2023

Crossrefs

Cf. A000984, A000989, A053824, A112765.

Sequence in context: A317756 A039998 A316089 * A175921 A303828 A025451

Adjacent sequences: A000996 A000997 A000998 * A001000 A001001 A001002

Keyword

nonn,easy

Author

N. J. A. Sloane, R. K. Guy

Extensions

More terms from Michael Somos, Jun 27 2002

Status

approved