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

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, 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, 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

Offset: 1

Author

Artur Jasinski, Apr 11 2008

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Wikipedia, Legendre's formula

Crossrefs

Prime equivalent of Kempner numbers A002034.

For quotients p!/n see A139170.

For indices of primes in this sequence see A139169.

Cf. A082672, A089085, A089130, A117141, A007749, A139056-A139066, A139068, A137390, A139070-A139075, A139148-A139157, A139159, A139160-A139166, A139089, A139168-A139170.

Programs

Maple

f:= proc(n) local F,m,Q,E,p;
  F:= ifactors(n)[2];
  m:= nops(F);
  Q:= map(t -> t[1],F);
  E:= map(t -> t[2],F);
  p:= max(Q)-1;
  do
    p:= nextprime(p);
    if andmap(i -> add(floor(p/Q[i]^j),j=1..floor(log[Q[i]](p))) >= E[i], [$1..m]) then return p fi;
  od
end proc:
f(1):= 2:
map(f, [$1..100]); # Robert Israel, Mar 07 2018

Mathematica

a = {}; Do[m = 1; While[ ! IntegerQ[Prime[m]!/n], m++ ]; AppendTo[a, Prime[m]], {n, 1, 100}]; a

PARI

a(n) = forprime(p=2,, if (!(p! % n), return (p))); \\ Michel Marcus, Mar 08 2018