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

1,1

Comments

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

Numbers k such that A356631(k) = 1.

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A001414, A001597, A025281, A356631.

Sequence in context: A272588 A272228 A051243 * A100411 A065405 A279627

Adjacent sequences: A356643 A356644 A356645 * A356647 A356648 A356649

Keyword

nonn

Author

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

Extensions

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

Status

approved