A356646 - OEIS

%I #20 Aug 30 2022 22:08:32

%S 4,8,27,31,575,669,1201,2505,4784,7618,35710,65005,166422,870062,

%T 994086,1105670,1209538,2140133,3020610,9147713,9404277,14492743,

%U 16792162,18566766,19445469,21264479,46483343,109424090,292374429,293351547,362681674,399576585,450622855

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

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

%C Numbers k such that A356631(k) = 1.

%H Chai Wah Wu, <a href="/A356646/b356646.txt">Table of n, a(n) for n = 1..38</a>

%e 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 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 for i from 1 while count < 27 do

%p   s:= s+spf(i);

%p   if ispow(s) then

%p     count:= count+1; R:= R,i;

%p   fi

%p od:

%p R;

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

%o (Python)

%o from itertools import count, islice, accumulate

%o from math import prod

%o from sympy import perfect_power, factorint

%o def A356646_gen(): # generator of terms

%o     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_list = list(islice(A356646_gen(),10)) # _Chai Wah Wu_, Aug 28 2022

%Y Cf. A001414, A001597, A025281, A356631.

%K nonn

%O 1,1

%A _J. M. Bergot_ and _Robert Israel_, Aug 19 2022

%E a(28)-a(33) from _Chai Wah Wu_, Aug 28 2022