A377821 -id:A377821 - cp's OEIS frontend

A377820 Powerful numbers that have a single odd exponent in their prime factorization.

8, 27, 32, 72, 108, 125, 128, 200, 243, 288, 343, 392, 432, 500, 512, 648, 675, 800, 968, 972, 1125, 1152, 1323, 1331, 1352, 1372, 1568, 1728, 1800, 2000, 2048, 2187, 2197, 2312, 2592, 2700, 2888, 3087, 3125, 3200, 3267, 3528, 3872, 3888, 4232, 4500, 4563, 4608, 4913, 5000

Offset: 1

Amiram Eldar, Nov 09 2024

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

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

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

Intersection of A001694 and A229125.
Intersection of A000037 and A377821.
Cf. A013661, A085541, A246551, A350389, A370786.

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 &]]

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

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

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

Original entry on oeis.org

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

A377820 Powerful numbers that have a single odd exponent in their prime factorization.

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