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

1,5

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A129768 A210715 A072926 * A345163 A369728 A369986

Adjacent sequences: A080084 A080085 A080086 * A080088 A080089 A080090

Keyword

nonn

Author

Paul D. Hanna, Jan 26 2003

Status

approved