A356646 Numbers k such that the integer log of k! is a perfect power.

4, 8, 27, 31, 575, 669, 1201, 2505, 4784, 7618, 35710, 65005, 166422, 870062, 994086, 1105670, 1209538, 2140133, 3020610, 9147713, 9404277, 14492743, 16792162, 18566766, 19445469, 21264479, 46483343, 109424090, 292374429, 293351547, 362681674, 399576585, 450622855

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 19 2022

Keywords

• nonn

Comments

Numbers k such that A025281(k) is a perfect power.

Numbers k such that A356631(k) = 1.

Examples

			a(2) = 8 because the integer log of 8! = 2^7 * 3^2 * 5 * 7 is 2*7 + 3*2 + 5 + 7 = 32 = 2^5 is a perfect power.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..38

Crossrefs

Cf. A001414, A001597, A025281, A356631.

Programs

Maple

spf:= proc(n) local t; add(t[1]*t[2],t=ifactors(n)[2]) end proc:ispow:= proc(n) igcd(map(t -> t[2], ifactors(n)[2]))>1 end proc:s:= 0: R:= NULL: count:= 0:
for i from 1 while count < 27 do
  s:= s+spf(i);
  if ispow(s) then
    count:= count+1; R:= R,i;
  fi
od:
R;

Mathematica

Select[Range[8000], GCD @@ FactorInteger[Plus @@ Times @@@ FactorInteger[#!]][[;; , 2]] > 1 &] (* Amiram Eldar, Aug 26 2022 *)

Python

from itertools import count, islice, accumulate
from math import prod
from sympy import perfect_power, factorint
def A356646_gen(): # generator of terms
    return (a+2 for a, b in enumerate(accumulate(sum(prod(d) for d in factorint(n).items()) for n in count(2))) if perfect_power(b))
A356646_list = list(islice(A356646_gen(),10)) # Chai Wah Wu, Aug 28 2022

Extensions

a(28)-a(33) from Chai Wah Wu, Aug 28 2022