A377847 - OEIS

%I #9 Nov 10 2024 05:41:04

%S 8,16,27,32,64,72,81,108,125,128,144,200,243,256,288,324,343,392,400,

%T 500,512,576,625,675,729,784,800,968,972,1024,1125,1152,1323,1331,

%U 1352,1372,1568,1600,1800,1936,2025,2048,2187,2197,2304,2312,2401,2500,2700,2704,2888,2916

%N Powerful numbers that are divisible by the cube of a single prime.

%C Numbers whose prime factorization contains one exponent that equals 3, and all the others, if they exist, are equal to 2.

%H Amiram Eldar, <a href="/A377847/b377847.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) = (15/Pi^2) * Sum_{p prime} 1/((p-1)*(p^2+1))) = 0.40258439321745859629... .

%t q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, e[[-1]] > 2 && (Length[e] == 1 || e[[-2]] == 2)]; With[{max = 3000}, Select[Union@ Flatten@Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], q]]

%o (PARI) is(k) = if(k == 1, 0, my(e = vecsort(factor(k)[, 2])); e[1] > 1 && e[#e] > 2 && (#e == 1 || e[#e - 1] == 2));

%Y Subsequence of A001694, A320966 and A377846.

%Y A030078 is a subsequence.

%Y Cf. A082020.

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Nov 09 2024