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

0,25

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

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 *)

Prog

(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
(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

Crossrefs

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

Cf. also A066874, A282568, A283550.

Sequence in context: A305558 A341624 A341620 * A065205 A036272 A257403

Adjacent sequences: A097795 A097796 A097797 * A097799 A097800 A097801

Keyword

nonn

Author

Reinhard Zumkeller, Aug 25 2004

Extensions

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

Status

approved