cp's OEIS Frontend

A334405 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.

%I A334405 #28 Mar 19 2024 13:53:49
%S A334405 6,28,36,60,84,90,120,156,210,216,240,252,270,300,312,330,336,352,396,
%T A334405 420,468,480,496,504,540,546,552,576,588,594,600,616,624,630,648,660,
%U A334405 672,714,720,756,760,780,784,792,816,840,864,888,900,924,960,972,1000
%N A334405 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.
%C A334405 Includes all the perfect numbers (A000396).
%C A334405 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
%H A334405 Amiram Eldar, <a href="/A334405/b334405.txt">Table of n, a(n) for n = 1..1650</a>
%H A334405 Tim McCormack and Joshua Zelinsky, <a href="https://arxiv.org/abs/2312.11661">Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba's Function</a>, arXiv:2312.11661 [math.NT], 2023. Mentions this sequence.
%F A334405 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.
%t A334405 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]
%Y A334405 Subsequence of A005835.
%Y A334405 A000396 is a subsequence.
%K A334405 nonn
%O A334405 1,1
%A A334405 _Amiram Eldar_, Apr 27 2020