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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 27 2020
STATUS
approved