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Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.
6

%I #22 Sep 28 2023 02:06:14

%S 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,

%T 25,25,26,31,32,33,33,35,37,38,39,40,41,43,44,46,46,47,47,51,53,55,55,

%U 56,57,58,62,63,64,65,66,68,69,69,71,75,76,76,77,81,82,84,84,86,87,89,90

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

%C 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

%H Robert Israel, <a href="/A080087/b080087.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = A112765(A039716(n)). - _Michel Marcus_, Sep 28 2023

%p R:= NULL: v:= 0: p:= 0:

%p for i from 1 to 100 do

%p q:= p;

%p p:= nextprime(p);

%p v:= v + add(1+padic:-ordp(x,5), x = 1+floor(q/5) .. floor(p/5));

%p R:= R,v;

%p od:

%p R; # _Robert Israel_, Sep 27 2023

%t 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 *)

%o (PARI) a(n) = valuation(prime(n)!, 5); \\ _Michel Marcus_, Jan 15 2015

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

%K nonn

%O 1,5

%A _Paul D. Hanna_, Jan 26 2003