%I #21 May 28 2023 20:49:11

%S 1,2,4,8,9,16,18,25,27,32,36,49,50,54,64,72,81,98,100,108,121,125,128,

%T 144,162,169,196,200,216,225,242,243,250,256,288,289,324,338,343,361,

%U 392,400,432,441,450,484,486,500,512,529,576,578,625,648,675,676,686

%N Numbers that are powerful in Gaussian integers.

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

%C The even powerful numbers divided by 4. - _Amiram Eldar_, May 28 2023

%H Amiram Eldar, <a href="/A335851/b335851.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GaussianInteger.html">Gaussian Integer</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Gaussian integer</a>.

%F 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...

%e 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.

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

%Y Disjoint union of A001694 and 2 * A062739.

%Y Cf. A082695.

%K nonn

%O 1,2

%A _Amiram Eldar_, Jun 26 2020