A080087 Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.

0, 0, 1, 1, 2, 2, 3, 3, 4, 6, 7, 8, 9, 9, 10, 12, 13, 14, 15, 16, 16, 18, 19, 20, 22, 24, 24, 25, 25, 26, 31, 32, 33, 33, 35, 37, 38, 39, 40, 41, 43, 44, 46, 46, 47, 47, 51, 53, 55, 55, 56, 57, 58, 62, 63, 64, 65, 66, 68, 69, 69, 71, 75, 76, 76, 77, 81, 82, 84, 84, 86, 87, 89, 90

Offset: 1

Author

Paul D. Hanna, Jan 26 2003

Keywords

• nonn

Comments

Highest power of 5 dividing prime(n)! = A039716(n), or also the number of trailing end 0's in A039716(n). - Lekraj Beedassy, Oct 31 2010

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A080084, A080085, A080086, A039716, A112765, A027868.

Programs

Maple

R:= NULL: v:= 0: p:= 0:
for i from 1 to 100 do
   q:= p;
   p:= nextprime(p);
   v:= v + add(1+padic:-ordp(x,5), x = 1+floor(q/5) .. floor(p/5));
   R:= R,v;
od:
R; # Robert Israel, Sep 27 2023

Mathematica

lst={};Do[p=Prime[n];s=0;While[p>1,p=IntegerPart[p/5];s+=p;];AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 28 2009 *)

PARI

a(n) = valuation(prime(n)!, 5); \\ Michel Marcus, Jan 15 2015

Formula

a(n) = Sum_{k=1..L} floor(prime(n)/5^k), where L = log(p_n)/log(5).

a(n) = A112765(A039716(n)). - Michel Marcus, Sep 28 2023