A300794 -id:A300794 - cp's OEIS frontend

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

24, 36, 48, 60, 84, 90, 96, 108, 120, 144, 168, 180, 216, 240, 264, 288, 300, 336, 360, 408, 420, 480, 540, 576, 588, 600, 660, 720, 780, 840, 924, 960, 1008, 1080, 1140, 1200, 1260, 1320, 1380, 1428, 1440, 1500, 1560, 1620, 1680, 1920, 1980, 2040, 2100, 2280

Offset: 1

Amiram Eldar, Mar 09 2019

nonn

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).

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.

			a(1) = 24 = 12 + 12 (one way);
a(2) = 36 = 12 + 24 = 18 + 18 (2 ways);
a(3) = 48 = 12 + 36 = 18 + 30 = 24 + 24 (3 ways);
a(4) = 60 = 12 + 48 = 18 + 42 = 20 + 40 = 24 + 36 = 30 + 30 (5 ways).

Eric A. Weiss, ed., A Computer Science Reader: Selections from ABACUS, Springer Science & Business Media, New York, 1988, p. 336.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Amiram Eldar, Table of n, a(n), record values for n = 1..1000
Richard V. Andree, Computer-Assisted Problem Solving, ABACUS, Vol. 2, No. 3 (Spring 1985), pp. 61-71 (reprinted in Weiss's book).

Cf. A005101, A048242, A283550, A300794.

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

cp's OEIS Frontend

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

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