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A361177 Exponentially powerful numbers: numbers whose exponents in their canonical prime factorization are all powerful numbers (A001694).

%I A361177 #10 Mar 04 2023 08:56:24
%S A361177 1,2,3,5,6,7,10,11,13,14,15,16,17,19,21,22,23,26,29,30,31,33,34,35,37,
%T A361177 38,39,41,42,43,46,47,48,51,53,55,57,58,59,61,62,65,66,67,69,70,71,73,
%U A361177 74,77,78,79,80,81,82,83,85,86,87,89,91,93,94,95,97,101,102
%N A361177 Exponentially powerful numbers: numbers whose exponents in their canonical prime factorization are all powerful numbers (A001694).
%C A361177 First differs from it subsequence A197680 at n = 167: a(167) = 256 is not a term of A197680.
%C A361177 The asymptotic density of this sequence is Product_{p prime} ((1 - 1/p)*(1 + Sum_{i>=1} 1/p^A001694(i))) = 0.6427901996... .
%H A361177 Amiram Eldar, <a href="/A361177/b361177.txt">Table of n, a(n) for n = 1..10000</a>
%t A361177 powQ[n_] := n == 1 || Min[FactorInteger[n][[;; , 2]]] > 1; Select[Range[100], AllTrue[FactorInteger[#][[;;, 2]], powQ] &]
%o A361177 (PARI) ispow(n) = {n == 1 || vecmin(factor(n)[,2]) > 1; }
%o A361177 is(n) = {my(e = factor(n)[, 2]); if(n == 1, return(1)); for(i=1, #e, if(!ispow(e[i]), return(0))); 1;}
%Y A361177 Cf. A001694.
%Y A361177 Similar sequences: A197680, A209061, A138302, A268335.
%K A361177 nonn
%O A361177 1,2
%A A361177 _Amiram Eldar_, Mar 03 2023