A377820 - OEIS

%I #12 Nov 09 2024 16:17:33

%S 8,27,32,72,108,125,128,200,243,288,343,392,432,500,512,648,675,800,

%T 968,972,1125,1152,1323,1331,1352,1372,1568,1728,1800,2000,2048,2187,

%U 2197,2312,2592,2700,2888,3087,3125,3200,3267,3528,3872,3888,4232,4500,4563,4608,4913,5000

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

%C First differs from A370786 at n = 124: A370786(124) = 27000 = 2^3 * 3^3 * 5*3 is not a term of this sequence.

%C Powerful numbers k such that A350389(k) is a prime power with an odd exponent (A246551).

%H Amiram Eldar, <a href="/A377820/b377820.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Pow#powerful">Index entries for sequences related to powerful numbers</a>.

%F Sum_{n>=1} 1/a(n) = zeta(2) * P(3) = A013661 * A085541 = 0.28747301899596333866... .

%t 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 (PARI) is(k) = if(k == 1, 0, my(e = factor(k)[, 2]); vecmin(e) > 1 && #select(x -> (x%2), e) == 1);

%Y Intersection of A001694 and A229125.

%Y Intersection of A000037 and A377821.

%Y Cf. A013661, A085541, A246551, A350389, A370786.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Nov 09 2024