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A335851
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Numbers that are powerful in Gaussian integers.
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5
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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
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OFFSET
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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...
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EXAMPLE
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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.
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MATHEMATICA
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gaussPowerQ[n_] := AllTrue[FactorInteger[n, GaussianIntegers -> True], Abs[First[#]] == 1 || Last[#] > 1 &]; Select[Range[1000], gaussPowerQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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