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A342399
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Unitary pseudoperfect numbers k such that no subset of the nontrivial unitary divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} adds up to k.
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1
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3510, 3770, 5670, 5810, 6790, 7630, 7910, 9590, 9730, 544310, 740870, 2070970, 4017310, 4095190, 5368510, 5569690, 5762330, 5838770, 5855290, 5856130, 5887630, 5902470, 5985770, 6006070, 6039530, 6075370, 6083630, 6181210, 6259610, 6471290, 7038710, 7065730, 7285390
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OFFSET
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1,1
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COMMENTS
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Numbers that are the sum of a proper subset of their aliquot unitary divisors but are not the sum of any subset of their nontrivial unitary divisors.
The unitary perfect numbers (A002827) which are a subset of the unitary pseudoperfect numbers (A293188) are excluded from this sequence since otherwise they would all be trivial terms: if k is a unitary perfect number then the sum of the divisors {d|k : 1 < d < k, gcd(d, k/d) = 1} is k-1, so any subset of them has a sum smaller than k.
The unitary pseudoperfect numbers are thus a disjoint union of the unitary perfect numbers, this sequence and A342398.
The unitary abundant numbers (A034683) are a disjoint union of the unitary weird numbers (A064114), this sequence and A342398.
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LINKS
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EXAMPLE
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3510 is a term since it is a unitary pseudoperfect number, 3510 = 1 + 2 + 5 + 13 + 27 + 54 + 65 + 130 + 135 + 270 + 351 + 702 + 1755, and the set of nontrivial unitary divisors of 3510, {d|3510 : 1 < d < 3510, gcd(d, 3510/d) = 1} = {2, 5, 10, 13, 26, 27, 54, 65, 130, 135, 270, 351, 702, 1755}, has no subset that adds up to 3510.
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MATHEMATICA
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q[n_] := Module[{d = Most @ Select[Divisors[n], CoprimeQ[#, n/#] &], x}, Plus @@ d > n && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, Length[d]}], {x, 0, n}], n] > 0 && SeriesCoefficient[Series[Product[1 + x^d[[i]], {i, 2, Length[d]}], {x, 0, n}], n] == 0]; Select[Range[10^4], q]
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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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