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

%I A080087 #22 Sep 28 2023 02:06:14
%S A080087 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 A080087 25,25,26,31,32,33,33,35,37,38,39,40,41,43,44,46,46,47,47,51,53,55,55,
%U A080087 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 A080087 Number of factors of 5 in the factorial of the n-th prime, counted with multiplicity.
%C A080087 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 A080087 Robert Israel, <a href="/A080087/b080087.txt">Table of n, a(n) for n = 1..10000</a>
%F A080087 a(n) = Sum_{k=1..L} floor(prime(n)/5^k), where L = log(p_n)/log(5).
%F A080087 a(n) = A112765(A039716(n)). - _Michel Marcus_, Sep 28 2023
%p A080087 R:= NULL: v:= 0: p:= 0:
%p A080087 for i from 1 to 100 do
%p A080087    q:= p;
%p A080087    p:= nextprime(p);
%p A080087    v:= v + add(1+padic:-ordp(x,5), x = 1+floor(q/5) .. floor(p/5));
%p A080087    R:= R,v;
%p A080087 od:
%p A080087 R; # _Robert Israel_, Sep 27 2023
%t A080087 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 A080087 (PARI) a(n) = valuation(prime(n)!, 5); \\ _Michel Marcus_, Jan 15 2015
%Y A080087 Cf. A080084, A080085, A080086, A039716, A112765, A027868.
%K A080087 nonn
%O A080087 1,5
%A A080087 _Paul D. Hanna_, Jan 26 2003