A000999 - OEIS

%I #23 Mar 07 2023 02:35:27

%S 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,

%T 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,

%U 3,3,2,2,2,3,3,1,1,1,2,2,1,1,1,2,2,1,1

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

%H T. D. Noe, <a href="/A000999/b000999.txt">Table of n, a(n) for n = 0..1000</a>

%H E. E. Kummer, <a href="https://doi.org/10.1515/crll.1852.44.93">Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen</a>, Journal für die reine und angewandte Mathematik, Vol. 44 (1852), pp. 93-146; <a href="https://eudml.org/doc/147500">alternative link</a>.

%H Dorel Miheţ, <a href="https://doi.org/10.1007/s12045-010-0123-4">Legendre's and Kummer's theorems again</a>, Resonance, Vol. 15, No. 12 (2010), pp. 1111-1121; <a href="https://www.ias.ac.in/public/Volumes/reso/015/12/1111-1121.pdf">alternative link</a>.

%H Armin Straub, Victor H. Moll and Tewodros Amdeberhan, <a href="https://eudml.org/doc/278348">The p-adic valuation of k-central binomial coefficients</a>, Acta Arithmetica, Vol. 140, No. 1 (2009), pp. 31-42.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kummer&#39;s_theorem">Kummer's theorem</a>.

%F From _Amiram Eldar_, Feb 12 2021: (Start)

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

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

%t Table[IntegerExponent[Binomial[2*n, n], 5], {n, 0, 100}] (* _T. D. Noe_, Jun 21 2012 *)

%o (PARI) a(n)=if(n<0,0,valuation(binomial(2*n,n),5))

%o (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

%Y Cf. A000984, A000989, A053824, A112765.

%K nonn,easy

%O 0,14

%A _N. J. A. Sloane_, _R. K. Guy_

%E More terms from _Michael Somos_, Jun 27 2002