This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A339343 #18 May 08 2021 06:33:24
%S A339343 20,88,104,272,304,350,368,464,572,650,1184,1312,1376,1504,1696,1888,
%T A339343 1952,3770,4288,4544,4672,5056,5312,5696,5704,5810,6208,6464,6592,
%U A339343 6790,6808,6848,6976,7144,7232,7630,7910,8024,8056,9590,9730,10744,11096,11288,13192
%N A339343 Abundant pseudoperfect numbers k such that no subset of the nontrivial divisors {d|k : 1 < d < k} sums to k.
%C A339343 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.
%C A339343 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.
%C A339343 The pseudoperfect numbers are thus a disjoint union of the perfect numbers, this sequence, and A136446.
%C A339343 The abundant numbers (A005101) are a disjoint union of the weird numbers (A006037), this sequence, and A136446.
%C A339343 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.
%C A339343 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).
%C A339343 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.
%C A339343 a(157) = A122036(1) = 351351 is the least (and currently the only known) odd term.
%H A339343 Amiram Eldar, <a href="/A339343/b339343.txt">Table of n, a(n) for n = 1..10000</a>
%e A339343 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.
%t A339343 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]
%Y A339343 Cf. A000203, A000396, A000668, A005100, A005101, A005835, A006037, A033880, A070824, A088831, A122036, A136446, A308710.
%Y A339343 Subsequence of A006036.
%Y A339343 A122036 is a subsequence.
%K A339343 nonn
%O A339343 1,1
%A A339343 _Amiram Eldar_, Nov 30 2020