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
Keywords
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
Programs
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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 *)
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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
Comments