A334405 - OEIS

%I #28 Mar 19 2024 13:53:49

%S 6,28,36,60,84,90,120,156,210,216,240,252,270,300,312,330,336,352,396,

%T 420,468,480,496,504,540,546,552,576,588,594,600,616,624,630,648,660,

%U 672,714,720,756,760,780,784,792,816,840,864,888,900,924,960,972,1000

%N 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 Includes all the perfect numbers (A000396).

%C 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 Amiram Eldar, <a href="/A334405/b334405.txt">Table of n, a(n) for n = 1..1650</a>

%H 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 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 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 Subsequence of A005835.

%Y A000396 is a subsequence.

%K nonn

%O 1,1

%A _Amiram Eldar_, Apr 27 2020