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

%I A356646 #20 Aug 30 2022 22:08:32
%S A356646 4,8,27,31,575,669,1201,2505,4784,7618,35710,65005,166422,870062,
%T A356646 994086,1105670,1209538,2140133,3020610,9147713,9404277,14492743,
%U A356646 16792162,18566766,19445469,21264479,46483343,109424090,292374429,293351547,362681674,399576585,450622855
%N A356646 Numbers k such that the integer log of k! is a perfect power.
%C A356646 Numbers k such that A025281(k) is a perfect power.
%C A356646 Numbers k such that A356631(k) = 1.
%H A356646 Chai Wah Wu, <a href="/A356646/b356646.txt">Table of n, a(n) for n = 1..38</a>
%e A356646 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.
%p A356646 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:
%p A356646 for i from 1 while count < 27 do
%p A356646   s:= s+spf(i);
%p A356646   if ispow(s) then
%p A356646     count:= count+1; R:= R,i;
%p A356646   fi
%p A356646 od:
%p A356646 R;
%t A356646 Select[Range[8000], GCD @@ FactorInteger[Plus @@ Times @@@ FactorInteger[#!]][[;; , 2]] > 1 &] (* _Amiram Eldar_, Aug 26 2022 *)
%o A356646 (Python)
%o A356646 from itertools import count, islice, accumulate
%o A356646 from math import prod
%o A356646 from sympy import perfect_power, factorint
%o A356646 def A356646_gen(): # generator of terms
%o A356646     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))
%o A356646 A356646_list = list(islice(A356646_gen(),10)) # _Chai Wah Wu_, Aug 28 2022
%Y A356646 Cf. A001414, A001597, A025281, A356631.
%K A356646 nonn
%O A356646 1,1
%A A356646 _J. M. Bergot_ and _Robert Israel_, Aug 19 2022
%E A356646 a(28)-a(33) from _Chai Wah Wu_, Aug 28 2022