A377847 -id:A377847 - cp's OEIS frontend

A377846 Powerful numbers that are not divisible by the cubes of more than one distinct prime.

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 441, 484, 500, 512, 529, 576, 625, 675, 676, 729, 784, 800, 841, 900, 961, 968, 972, 1024, 1089, 1125, 1152, 1156, 1225

Offset: 1

Amiram Eldar, Nov 09 2024

Subsequence of A377821 and first differs from it at n = 33: A377821(33) = 432 = 2^4 * 3^3 is not a term of this sequence.

Numbers whose prime factorization has exponents that are all larger than 1 and no more than one exponent is larger than 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Complement of A376936 within A001694.
Subsequence of A377821.
Subsequences: A143610, A377847.
Cf. A082020.

q[n_] := Module[{e = Sort[FactorInteger[n][[;; , 2]]]}, Length[e] == 1 || e[[-2]] == 2]; With[{max = 1300}, Select[Union@ Flatten@ Table[i^2 * j^3, {j, 1, max^(1/3)}, {i, 1, Sqrt[max/j^3]}], # == 1 || q[#] &]]

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

Sum_{n>=1} 1/a(n) = (15/Pi^2) * (1 + Sum_{p prime} 1/((p-1)*(p^2+1))) = 1.92240214785252516795... .

cp's OEIS Frontend

A377846 Powerful numbers that are not divisible by the cubes of more than one distinct prime.

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