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A309568
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Bi-unitary k-hyperperfect numbers: numbers m such that m = 1 + k * (bsigma(m) - m - 1) where bsigma(m) is the sum of bi-unitary divisors of m (A188999) and k >= 1 is an integer.
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1
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6, 21, 52, 60, 90, 301, 657, 697, 1333, 1909, 2041, 2133, 3901, 15025, 24601, 26977, 96361, 130153, 163201, 176661, 250321, 275833, 296341, 389593, 486877, 495529, 542413, 808861, 1005421, 1005649, 1055833, 1063141, 1232053, 1246417, 1284121, 1357741, 1403221
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OFFSET
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1,1
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COMMENTS
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The only bi-unitary 1-hyperperfect numbers are 6, 60, and 90 (the bi-unitary perfect numbers).
The corresponding k values are 1, 2, 3, 1, 1, 6, 8, 12, 18, 18, 12, 2, 30, 24, 60, 48, 132, 132, 192, 2, 168, 108, 66, 252, 78, 132, 342, 366, 390, 168, 348, 282, 498, 552, 540, 30, 546, ...
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LINKS
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EXAMPLE
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21 is in the sequence since bsigma(21) = 32 and 21 = 1 + 2 * (32 - 21 - 1).
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MATHEMATICA
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fun[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p-1), (p^(e+1)-1)/(p-1)-p^(e/2)]; bsigma[1] = 1; bsigma[n_] := Times @@ (fun @@@ FactorInteger[n]); hpnQ[n_] := (c = bsigma[n]-n-1) > 0 && Divisible[n-1, c]; Select[Range[10^5], hpnQ]
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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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