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A139171 a(n) = smallest prime number p such that p!/n is an integer.

%I A139171 #15 Mar 08 2018 03:01:43
%S A139171 2,2,3,5,5,3,7,5,7,5,11,5,13,7,5,7,17,7,19,5,7,11,23,5,11,13,11,7,29,
%T A139171 5,31,11,11,17,7,7,37,19,13,5,41,7,43,11,7,23,47,7,17,11,17,13,53,11,
%U A139171 11,7,19,29,59,5,61,31,7,11,13,11,67,17,23,7,71,7,73,37,11,19,11,13,79,7,11
%N A139171 a(n) = smallest prime number p such that p!/n is an integer.
%H A139171 Robert Israel, <a href="/A139171/b139171.txt">Table of n, a(n) for n = 1..10000</a>
%H A139171 Wikipedia, <a href="https://en.wikipedia.org/wiki/Legendre%27s_formula">Legendre's formula</a>
%p A139171 f:= proc(n) local F,m,Q,E,p;
%p A139171   F:= ifactors(n)[2];
%p A139171   m:= nops(F);
%p A139171   Q:= map(t -> t[1],F);
%p A139171   E:= map(t -> t[2],F);
%p A139171   p:= max(Q)-1;
%p A139171   do
%p A139171     p:= nextprime(p);
%p A139171     if andmap(i -> add(floor(p/Q[i]^j),j=1..floor(log[Q[i]](p))) >= E[i], [$1..m]) then return p fi;
%p A139171   od
%p A139171 end proc:
%p A139171 f(1):= 2:
%p A139171 map(f, [$1..100]); # _Robert Israel_, Mar 07 2018
%t A139171 a = {}; Do[m = 1; While[ ! IntegerQ[Prime[m]!/n], m++ ]; AppendTo[a, Prime[m]], {n, 1, 100}]; a
%o A139171 (PARI) a(n) = forprime(p=2,, if (!(p! % n), return (p))); \\ _Michel Marcus_, Mar 08 2018
%Y A139171 Prime equivalent of Kempner numbers A002034.
%Y A139171 For quotients p!/n see A139170.
%Y A139171 For indices of primes in this sequence see A139169.
%Y A139171 Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160-A139166, A139089, A139168-A139170.
%K A139171 nonn,look
%O A139171 1,1
%A A139171 _Artur Jasinski_, Apr 11 2008