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A334405
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Pseudoperfect numbers k such that there is a subset of divisors of k whose sum is 2*k and for each d in this subset k/d is also in it.
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4
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6, 28, 36, 60, 84, 90, 120, 156, 210, 216, 240, 252, 270, 300, 312, 330, 336, 352, 396, 420, 468, 480, 496, 504, 540, 546, 552, 576, 588, 594, 600, 616, 624, 630, 648, 660, 672, 714, 720, 756, 760, 780, 784, 792, 816, 840, 864, 888, 900, 924, 960, 972, 1000
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OFFSET
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1,1
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COMMENTS
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Includes all the perfect numbers (A000396).
The McCormack and Zelinsky preprint shows that no terms are 2 (mod 3), and also that no terms are 3 (mod 4). That paper also asks if there are infinitely many odd terms. Empirically, odd terms are much rarer than even terms. - Joshua Zelinsky, Feb 28 2024
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LINKS
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FORMULA
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36 is a term since {1, 2, 3, 12, 18, 36} is a subset of its divisors whose sum is 72 = 2 * 36, and for each divisor d in this subset 36/d is also in it: 1 * 36 = 2 * 18 = 3 * 12 = 36.
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MATHEMATICA
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seqQ[n_] := Module[{d = Divisors[n]}, nd = Length[d]; divpairs = If[EvenQ[nd], d[[1 ;; nd/2]] + d[[-1 ;; nd/2 + 1 ;; -1]], Join[d[[1 ;; (nd - 1)/2]] + d[[-1 ;; (nd + 3)/2 ;; -1]], {d[[(nd + 1)/2]]}]]; SeriesCoefficient[Series[Product[1 + x^divpairs[[i]], {i, Length[divpairs]}], {x, 0, 2*n}], 2*n] > 0]; Select[Range[1000], seqQ]
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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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