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A306776 Numbers that are the sum of two abundant numbers (not necessarily distinct) in a record number of ways.

%I A306776 #24 Mar 14 2019 04:19:52
%S A306776 24,36,48,60,84,90,96,108,120,144,168,180,216,240,264,288,300,336,360,
%T A306776 408,420,480,540,576,588,600,660,720,780,840,924,960,1008,1080,1140,
%U A306776 1200,1260,1320,1380,1428,1440,1500,1560,1620,1680,1920,1980,2040,2100,2280
%N A306776 Numbers that are the sum of two abundant numbers (not necessarily distinct) in a record number of ways.
%C A306776 The record values of number of ways are 1, 2, 3, 5, 6, 7, 8, 9, 11, 13, 15, 17, ... (see link for more values).
%C A306776 According to Andree, Mr. James Jones of Moore, Oklahoma, has shown that 371280 can be expressed as a sum of two abundant numbers in more than 43000 different ways and that record-breaking values are likely to be multiples of 60. Indeed, except for the 19 terms 24, 36, 48, 84, 90, 96, 108, 144, 168, 216, 264, 288, 336, 408, 576, 588, 924, 1008, and 1428, apparently all the others are divisible by 60.
%D A306776 Eric A. Weiss, ed., A Computer Science Reader: Selections from ABACUS, Springer Science & Business Media, New York, 1988, p. 336.
%H A306776 Amiram Eldar, <a href="/A306776/b306776.txt">Table of n, a(n) for n = 1..1000</a>
%H A306776 Amiram Eldar, <a href="/A306776/a306776.txt">Table of n, a(n), record values for n = 1..1000</a>
%H A306776 Richard V. Andree, <a href="https://doi.org/10.1007/978-1-4419-8726-6_23">Computer-Assisted Problem Solving</a>, ABACUS, Vol. 2, No. 3 (Spring 1985), pp. 61-71 (reprinted in Weiss's book).
%e A306776 a(1) = 24 = 12 + 12 (one way);
%e A306776 a(2) = 36 = 12 + 24 = 18 + 18 (2 ways);
%e A306776 a(3) = 48 = 12 + 36 = 18 + 30 = 24 + 24 (3 ways);
%e A306776 a(4) = 60 = 12 + 48 = 18 + 42 = 20 + 40 = 24 + 36 = 30 + 30 (5 ways).
%t A306776 nm=1000; ab=Select[Range[nm], DivisorSigma[1,#] > 2# &]; f[n_] := Length[ IntegerPartitions[n, {2}, ab]]; s={}; fm=0; Do[f1 = f[n]; If[f1>fm, fm=f1; AppendTo[s,n]], {n, 1, nm}]; s
%Y A306776 Cf. A005101, A048242, A283550, A300794.
%K A306776 nonn
%O A306776 1,1
%A A306776 _Amiram Eldar_, Mar 09 2019