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

Author

Reinhard Zumkeller, Aug 25 2004

Keywords

• nonn

Comments

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

Links

• 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

Crossrefs

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

Cf. also A066874, A282568, A283550.

Programs

Magma

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

Mathematica

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 *)

PARI

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

Extensions

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