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A339343
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Abundant pseudoperfect numbers k such that no subset of the nontrivial divisors {d|k : 1 < d < k} sums to k.
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2
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20, 88, 104, 272, 304, 350, 368, 464, 572, 650, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 3770, 4288, 4544, 4672, 5056, 5312, 5696, 5704, 5810, 6208, 6464, 6592, 6790, 6808, 6848, 6976, 7144, 7232, 7630, 7910, 8024, 8056, 9590, 9730, 10744, 11096, 11288, 13192
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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 divisors but are not the sum of any subset of their nontrivial divisors.
The perfect numbers (A000396) which are a subset of the pseudoperfect numbers (A005835) are excluded from this sequence since otherwise they would all be trivial terms: if k is a perfect number then the sum of the divisors {d|k : 1 < d < k} is k-1, so any subset of them has a sum smaller than k.
The pseudoperfect numbers are thus a disjoint union of the perfect numbers, this sequence, and A136446.
The abundant numbers (A005101) are a disjoint union of the weird numbers (A006037), this sequence, and A136446.
All the terms are primitive pseudoperfect (A006036), since if k*m is a pseudoperfect number with k > 1, and m also pseudoperfect, then it is a sum of a subset of its divisors, all of which are multiples of k and therefore larger than 1.
This sequence is infinite. If p is an odd prime that is not a Mersenne prime (A000668), and k is the least number such that 2^k * p is an abundant number (A005101; i.e., the least k such that 2^(k+1) - 1 > p), then 2^k * p is a term (these are the nonperfect terms of A308710). If 2^k * p was not a term, then since it has only 2 odd divisors (1 and p), it would be equal to a sum of its even divisors (if 1 is not in the sum then p also cannot be in it). This would make 2^(k-1) * p also a pseudoperfect number, but by definition of k, 2^(k-1) * p is a deficient number (A005100).
If k is an even abundant number with abundance (A033880) 2, i.e., sigma(k) = A000203(k) = 2*k + 2, then k is a term.
a(157) = A122036(1) = 351351 is the least (and currently the only known) odd term.
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LINKS
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EXAMPLE
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20 is a term since it is a pseudoperfect number, 20 = 1 + 4 + 5 + 10, and the set of nontrivial divisors of 20, {d|20 : 1 < d < 20} = {2, 4, 5, 10}, has no subset that sums to 20.
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MATHEMATICA
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psQ[n_] := DivisorSigma[1, n] > 2*n && Module[{d = Most@Divisors[n], x}, 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[2000], psQ]
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CROSSREFS
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Cf. A000203, A000396, A000668, A005100, A005101, A005835, A006037, A033880, A070824, A088831, A122036, A136446, A308710.
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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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