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.

6, 28, 36, 60, 84, 90, 120, 156, 210, 216, 240, 252, 270, 300, 312, 330, 336, 352, 396, 420, 468, 480, 496, 504, 540, 546, 552, 576, 588, 594, 600, 616, 624, 630, 648, 660, 672, 714, 720, 756, 760, 780, 784, 792, 816, 840, 864, 888, 900, 924, 960, 972, 1000

Offset: 1

Author

Amiram Eldar, Apr 27 2020

Keywords

• nonn

Comments

Includes all the perfect numbers (A000396).

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1650
• Tim McCormack and Joshua Zelinsky, Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba's Function, arXiv:2312.11661 [math.NT], 2023. Mentions this sequence.

Crossrefs

Subsequence of A005835.

A000396 is a subsequence.

Programs

Mathematica

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]

Formula

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.