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

1,1

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.

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.

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]

Crossrefs

Subsequence of A005835.

A000396 is a subsequence.

Sequence in context: A272971 A117948 A252234 * A376997 A242344 A344588

Adjacent sequences: A334402 A334403 A334404 * A334406 A334407 A334408

Keyword

nonn

Author

Amiram Eldar, Apr 27 2020

Status

approved