OFFSET
1,2
COMMENTS
Numbers that are "squarefree" when they are factorized into factors of the form p^(k!), where p is a prime and k >= 1, a factorization that is done using the factorial-base representation of the exponents in the prime factorization (see A376885 for more details). Each factor p^(k!) has a multiplicity 1.
The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^2 + (1 - 1/p) * (Sum_{k>=2} 1/p^A059590(k))) = 0.93973112474919498992... .
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
MATHEMATICA
expQ[n_] := expQ[n] = Module[{k = n, m = 2, r, s = 1}, While[{k, r} = QuotientRemainder[k, m]; k != 0 || r != 0, If[r > 1, s = 0; Break[]]; m++]; s == 1]; q[n_] := AllTrue[FactorInteger[n][[;; , 2]], expQ]; Select[Range[100], q]
PROG
(PARI) isexp(n) = {my(k = n, m = 2, r); while([k, r] = divrem(k, m); k != 0 || r != 0, if(r > 1, return(0)); m++); 1; }
is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1; }
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Oct 13 2024
STATUS
approved