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A376171 Powerful numbers whose prime factorization has an odd maximum exponent.

%I A376171 #11 Sep 14 2024 06:52:34
%S A376171 8,27,32,72,108,125,128,200,216,243,288,343,392,500,512,675,800,864,
%T A376171 968,972,1000,1125,1152,1323,1331,1352,1372,1568,1800,1944,2048,2187,
%U A376171 2197,2312,2592,2700,2744,2888,3087,3125,3200,3267,3375,3456,3528,3872,3888,4000
%N A376171 Powerful numbers whose prime factorization has an odd maximum exponent.
%C A376171 Subsequence of A102834 and first differs from it at n = 14: A102834(14) = 432 = 2^4 * 3^3 is not a term of this sequence.
%C A376171 Powerful numbers k such that A051903(k) is odd.
%C A376171 Equivalently, numbers whose prime factorization exponents are all larger than 1 and their maximum is odd. The maximum exponent in the prime factorization of 1 is considered to be A051903(1) = 0, and therefore 1 is not a term of this sequence.
%C A376171 The numbers of terms that do not exceed the 10^k-powerful number (A376092(k)), for k = 1, 2, ..., are 3, 40, 416, 4255, 42829, 429393, 4299797, 43022803, ... . Apparently, the asymptotic density of this sequence within the powerful numbers (A001694) exists and approximately equals 0.43.
%H A376171 Amiram Eldar, <a href="/A376171/b376171.txt">Table of n, a(n) for n = 1..10000</a>
%H A376171 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A376171 Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - Sum_{k>=2} (-1)^k * s(k) = 0.29116340833243888282..., where s(k) = Product_{p prime} (1 + Sum_{i=2..k} 1/p^i).
%t A376171 seq[lim_] := Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, Surd[lim, 3]}, {i, 1, Sqrt[lim/j^3]}], # > 1 && OddQ[Max[FactorInteger[#][[;; , 2]]]] &]; seq[10^4]
%o A376171 (PARI) is(k) = {my(f = factor(k), e = f[,2]); #e && ispowerful(f) && vecmax(e) % 2;}
%Y A376171 Complement of A376170 within A001694.
%Y A376171 Intersection of A001694 and A376142.
%Y A376171 Subsequence of A102834.
%Y A376171 Subsequences: A030078, A050997, A079395, A092759, A138031, A179665, A335988 \ {1}.
%Y A376171 Cf. A051903, A082695, A376092.
%K A376171 nonn,easy
%O A376171 1,1
%A A376171 _Amiram Eldar_, Sep 13 2024