OFFSET
1,2
COMMENTS
Numbers that are "powerful" 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 that is larger than 1.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + Sum_{k>=1} 1/p^A255411(k)) = 1.07819745085315583226... .
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]; seq[lim_] := Module[{p = 2, s = {1}, emax, es}, While[(emax = Floor[Log[p, lim]]) > 3, es = Select[Range[0, emax], expQ]; s = Union[s, Select[Union[Flatten[Outer[Times, s, p^es]]], # <= lim &]]; p = NextPrime[p]]; s]; seq[4*10^6]
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,base
AUTHOR
Amiram Eldar, Oct 13 2024
STATUS
approved