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A377021
Numbers whose prime factorization has exponents that are all sums of distinct factorials (A059590, where 0! and 1! are not considered distinct).
4
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71
OFFSET
1,2
COMMENTS
First differs from its subsequence A046100 at n = 61: a(61) = 64 is not a term of A046100.
Numbers k such that A376885(k) = A376886(k).
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
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
Analogous to A005117.
Sequence in context: A178210 A013938 A339889 * A023809 A046100 A337138
KEYWORD
nonn,easy
AUTHOR
Amiram Eldar, Oct 13 2024
STATUS
approved