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A377820 Powerful numbers that have a single odd exponent in their prime factorization.

%I A377820 #12 Nov 09 2024 16:17:33
%S A377820 8,27,32,72,108,125,128,200,243,288,343,392,432,500,512,648,675,800,
%T A377820 968,972,1125,1152,1323,1331,1352,1372,1568,1728,1800,2000,2048,2187,
%U A377820 2197,2312,2592,2700,2888,3087,3125,3200,3267,3528,3872,3888,4232,4500,4563,4608,4913,5000
%N A377820 Powerful numbers that have a single odd exponent in their prime factorization.
%C A377820 First differs from A370786 at n = 124: A370786(124) = 27000 = 2^3 * 3^3 * 5*3 is not a term of this sequence.
%C A377820 Powerful numbers k such that A350389(k) is a prime power with an odd exponent (A246551).
%H A377820 Amiram Eldar, <a href="/A377820/b377820.txt">Table of n, a(n) for n = 1..10000</a>
%H A377820 <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.
%F A377820 Sum_{n>=1} 1/a(n) = zeta(2) * P(3) = A013661 * A085541 = 0.28747301899596333866... .
%t A377820 With[{max = 5000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # > 1 && Count[FactorInteger[#][[;; , 2]], _?OddQ] == 1 &]]
%o A377820 (PARI) is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x%2), e) == 1);
%Y A377820 Intersection of A001694 and A229125.
%Y A377820 Intersection of A000037 and A377821.
%Y A377820 Cf. A013661, A085541, A246551, A350389, A370786.
%K A377820 nonn,easy
%O A377820 1,1
%A A377820 _Amiram Eldar_, Nov 09 2024