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

1,1

Comments

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.

References

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

Links

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).

Example

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).

Mathematica

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

Crossrefs

Cf. A005101, A048242, A283550, A300794.

Sequence in context: A273088 A334674 A067766 * A300794 A327945 A078347

Adjacent sequences: A306773 A306774 A306775 * A306777 A306778 A306779

Keyword

nonn

Author

Amiram Eldar, Mar 09 2019

Status

approved