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A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).

%I A382065 #8 Mar 14 2025 21:17:12
%S A382065 1,2,3,4,5,6,7,9,10,11,12,13,14,15,17,18,19,20,21,22,23,25,26,28,29,
%T A382065 30,31,33,34,35,36,37,38,39,41,42,43,44,45,46,47,49,50,51,52,53,55,57,
%U A382065 58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79
%N A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).
%C A382065 First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.
%C A382065 First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.
%C A382065 All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8.
%C A382065 Subsequence of A382063 and first differs from it at n = 362: A382063(362) = 432 = 2^4 * 3^3 is not a term of this sequence.
%C A382065 The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A033950(k))) = 0.83493143539605138255... .
%C A382065 The relative density of this sequence within A382063 is the ratio between the densities of the two sequences: 0.997553... .
%H A382065 Amiram Eldar, <a href="/A382065/b382065.txt">Table of n, a(n) for n = 1..10000</a>
%t A382065 refQ[k_] := Divisible[k, DivisorSigma[0, k]]; q[k_] := AllTrue[FactorInteger[k][[;; , 2]], refQ]; Select[Range[100], q]
%o A382065 (PARI) isref(n) = !(n % numdiv(n));
%o A382065 isok(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isref(e[i]), return(0))); 1; }
%Y A382065 Cf. A033950, A344742.
%Y A382065 Subsequence of A382063.
%Y A382065 Subsequence: A004709.
%Y A382065 Similar sequences: A197680, A209061, A138302, A268335, A361177, A377019.
%K A382065 nonn,easy
%O A382065 1,2
%A A382065 _Amiram Eldar_, Mar 14 2025