A377847 Powerful numbers that are divisible by the cube of a single prime.

8, 16, 27, 32, 64, 72, 81, 108, 125, 128, 144, 200, 243, 256, 288, 324, 343, 392, 400, 500, 512, 576, 625, 675, 729, 784, 800, 968, 972, 1024, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1600, 1800, 1936, 2025, 2048, 2187, 2197, 2304, 2312, 2401, 2500, 2700, 2704, 2888, 2916

Offset

1,1

Comments

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

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Index entries for sequences related to powerful numbers.

Formula

Sum_{n>=1} 1/a(n) = (15/Pi^2) * Sum_{p prime} 1/((p-1)*(p^2+1))) = 0.40258439321745859629... .

Mathematica

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]]

Prog

(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));

Crossrefs

Subsequence of A001694, A320966 and A377846.

A030078 is a subsequence.

Cf. A082020.

Sequence in context: A349306 A245713 A320966 * A036966 A076467 A111231

Adjacent sequences: A377844 A377845 A377846 * A377849 A377852 A377853

Keyword

nonn,easy

Author

Amiram Eldar, Nov 09 2024

Status

approved