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

1,2

Comments

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.

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) * (1 + Sum_{p prime} 1/((p-1)*(p^2+1))) = 1.92240214785252516795... .

Mathematica

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[#] &]]

Prog

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

Crossrefs

Complement of A376936 within A001694.

Subsequence of A377821.

Subsequences: A143610, A377847.

Cf. A082020.

Sequence in context: A080366 A001694 A377821 * A317102 A157985 A001597

Adjacent sequences: A377843 A377844 A377845 * A377847 A377848 A377849

Keyword

nonn,easy

Author

Amiram Eldar, Nov 09 2024

Status

approved