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

Author

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

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A000984, A000989, A053824, A112765.

Programs

Mathematica

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

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

Formula

From Amiram Eldar, Feb 12 2021: (Start)

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

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

Extensions

More terms from Michael Somos, Jun 27 2002