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

1,2

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

Links

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

Eric Weisstein's World of Mathematics, Gaussian Integer.

Wikipedia, Gaussian integer.

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...

Example

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.

Mathematica

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

Crossrefs

Disjoint union of A001694 and 2 * A062739.

Cf. A082695.

Sequence in context: A048715 A336232 A242662 * A028982 A320137 A324525

Adjacent sequences: A335848 A335849 A335850 * A335852 A335853 A335854

Keyword

nonn

Author

Amiram Eldar, Jun 26 2020

Status

approved