A265310 Least positive k such that the product of divisors of n (A007955) divides k!.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 9, 13, 14, 10, 12, 17, 15, 19, 15, 14, 22, 23, 16, 15, 26, 15, 21, 29, 20, 31, 16, 22, 34, 14, 21, 37, 38, 26, 20, 41, 28, 43, 33, 15, 46, 47, 24, 21, 25, 34, 39, 53, 27, 22, 28, 38, 58, 59, 25, 61, 62, 21, 24, 26, 44, 67, 51, 46, 28, 71, 27, 73

Offset: 1

Author

Gionata Neri, Dec 06 2015

Keywords

• nonn

Comments

Conjecture: a(n) = n if and only if n is prime, 2*prime, 1, 8 or 9.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007955.

Programs

Maple

A265310:= proc(n) local F,f,tau,a,p,k;
      F:= ifactors(n)[2];
      tau:= mul(1+f[2],f=F);
      k:= 1;
      for f in F do
        a:= f[2]*tau/2;
        p:= f[1];
        while add(floor(k/p^j),j=1..ilog[p](k)) < a do k:= p*(1+floor(k/p)) od;
      od;
      k
end proc:
map(A265310, [$1..100]); # Robert Israel, Dec 07 2015

Mathematica

Table[k = 1; While[! Divisible[k!, Times @@ Divisors@ n], k++]; k, {n, 73}] (* Michael De Vlieger, Dec 06 2015 *)

PARI

a007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2));
a(n) = {k=1; while(k, if(k! % a007955(n)==0, return(k)); k++)}
vector(100, n, a(n)) \\ Altug Alkan, Dec 06 2015