A335851 Numbers that are powerful in Gaussian integers.

1, 2, 4, 8, 9, 16, 18, 25, 27, 32, 36, 49, 50, 54, 64, 72, 81, 98, 100, 108, 121, 125, 128, 144, 162, 169, 196, 200, 216, 225, 242, 243, 250, 256, 288, 289, 324, 338, 343, 361, 392, 400, 432, 441, 450, 484, 486, 500, 512, 529, 576, 578, 625, 648, 675, 676, 686

Offset: 1

Author

Amiram Eldar, Jun 26 2020

Keywords

• nonn

Comments

Numbers all of whose prime factors in Gaussian integers have multiplicity larger than 1.

The even powerful numbers divided by 4. - Amiram Eldar, May 28 2023

Examples

			2 is a term since 2 = -i * (1 + i)^2 in the ring of Gaussian integers. -i is a unit, and the multiplicity of its only Gaussian prime factor, 1 + i, is 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Gaussian Integer.
• Wikipedia, Gaussian integer.

Crossrefs

Disjoint union of A001694 and 2 * A062739.

Cf. A082695.

Programs

Mathematica

gaussPowerQ[n_] := AllTrue[FactorInteger[n, GaussianIntegers -> True], Abs[First[#]] == 1 || Last[#] > 1 &]; Select[Range[1000], gaussPowerQ]

Formula

Sum_{n>=1} 1/a(n) = (4/3) * Sum_{n>=1} 1/A001694(n) = 4*zeta(2)*zeta(3)/(3*zeta(6)) = (4/3) * A082695 = 2.591461...