cp's OEIS Frontend

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

%I A377021 #14 Mar 21 2025 02:23:44
%S A377021 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,
%T A377021 28,29,30,31,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,52,
%U A377021 53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71
%N A377021 Numbers whose prime factorization has exponents that are all sums of distinct factorials (A059590, where 0! and 1! are not considered distinct).
%C A377021 First differs from its subsequence A046100 at n = 61: a(61) = 64 is not a term of A046100.
%C A377021 Numbers k such that A376885(k) = A376886(k).
%C A377021 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.
%C A377021 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... .
%H A377021 Amiram Eldar, <a href="/A377021/b377021.txt">Table of n, a(n) for n = 1..10000</a>
%t A377021 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]
%o A377021 (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;}
%o A377021 is(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isexp(e[i]), return(0))); 1;}
%Y A377021 Cf. A000142, A046100, A059590, A376885, A376886, A377019, A377020, A377022.
%Y A377021 Analogous to A005117.
%K A377021 nonn,easy
%O A377021 1,2
%A A377021 _Amiram Eldar_, Oct 13 2024