A283550 -id:A283550 - cp's OEIS frontend

A097798 Number of partitions of n into abundant numbers.

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 0, 2, 0, 4, 0, 2, 0, 0, 0, 7, 0, 2, 0, 2, 0, 8, 0, 5, 0, 2, 0, 14, 0, 4, 0, 4, 0, 14, 0, 8, 0, 5, 0, 23, 0, 9, 0, 9, 0, 26, 0, 18, 0, 9, 0, 38, 0, 16, 0, 17, 0, 46, 0, 29, 0, 19, 0, 65, 0, 32, 0

Offset: 0

Reinhard Zumkeller, Aug 25 2004

nonn

n = 977 = 945 + 32 is the first prime for which sequence obtains a nonzero value, as a(977) = a(32) = 1. 945 is the first term in A005231. - Antti Karttunen, Sep 06 2018

a(n) = 0 for 496 values of n, the largest of which is 991 (see A283550). - David A. Corneth, Sep 08 2018

David A. Corneth, Table of n, a(n) for n = 0..10000 (a(1) through a(532) by Antti Karttunen)
David A. Corneth, PARI program
Eric Weisstein's World of Mathematics, Abundant Number
Eric Weisstein's World of Mathematics, Partition

Cf. A005101, A005231, A000041, A097800, A097797, A097796, A097795.

Cf. also A066874, A282568, A283550.

v:=[n:n in [1..100]| SumOfDivisors(n) gt 2*n]; [#RestrictedPartitions(n,Set(v)): n in [0..100]]; // Marius A. Burtea, Aug 02 2019

n = 100; d = Select[Range[n], DivisorSigma[1, #] > 2 # &]; CoefficientList[ Series[1/Product[1 - x^d[[i]], {i, 1, Length[d]} ], {x, 0, n}], x] (* Amiram Eldar, Aug 02 2019 *)

abundants_up_to_reversed(n) = { my(s = Set([])); for(k=1,n,if(sigma(k)>(2*k),s = setunion([k],s))); vecsort(s, ,4); };
partitions_into(n,parts,from=1) = if(!n,1,my(k = #parts, s=0); for(i=from,k,if(parts[i]<=n, s += partitions_into(n-parts[i],parts,i))); (s));
A097798(n) = partitions_into(n,abundants_up_to_reversed(n)); \\ Antti Karttunen, Sep 06 2018

PARI
```
\\ see Corneth link
```

a(0) = 1 prepended by David A. Corneth, Sep 08 2018

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

Original entry on oeis.org

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

A364728 Numbers that are not the sum of admirable numbers (not necessarily distinct).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 48, 49, 51, 53, 55, 57, 58, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101

Offset: 1

Amiram Eldar, Aug 05 2023

First differs from A053460 at n = 39.

Giovanni Resta found that 1003 is the largest number that is not a sum of admirable numbers.

Amiram Eldar, Table of n, a(n) for n = 1..504 (complete sequence)
Giovanni Resta, admirable numbers, Numbers Aplenty.

Cf. A053460, A111592.

Analogous sequence with abundant numbers: A283550.

admQ[n_] := (ab = DivisorSigma[1, n] - 2 n) > 0 && EvenQ[ab] && ab/2 < n && Divisible[n, ab/2];
With[{adm = Select[Range[1200], admQ]}, Position[Rest[CoefficientList[Series[Product[(1 + x^adm[[k]]), {k, 1, Length[adm]}], {x, 0, adm[[-1]]}], x]], 0] // Flatten]

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A097798 Number of partitions of n into abundant numbers.

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

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A364728 Numbers that are not the sum of admirable numbers (not necessarily distinct).

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Mathematica