A377021 Numbers whose prime factorization has exponents that are all sums of distinct factorials (A059590, where 0! and 1! are not considered distinct).

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

Author

Amiram Eldar, Oct 13 2024

Keywords

• nonn
• easy

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

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000142, A046100, A059590, A376885, A376886, A377019, A377020, A377022.

Analogous to A005117.

Programs

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]

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;}