%I #36 Sep 08 2022 08:45:14

%S 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,

%T 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,

%U 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

%N Number of partitions of n into abundant numbers.

%C 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

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

%H David A. Corneth, <a href="/A097798/b097798.txt">Table of n, a(n) for n = 0..10000</a> (a(1) through a(532) by Antti Karttunen)

%H David A. Corneth, <a href="/A097798/a097798_1.gp.txt">PARI program</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AbundantNumber.html">Abundant Number</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Partition.html">Partition</a>

%t 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 *)

%o (PARI)

%o abundants_up_to_reversed(n) = { my(s = Set([])); for(k=1,n,if(sigma(k)>(2*k),s = setunion([k],s))); vecsort(s, ,4); };

%o 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));

%o A097798(n) = partitions_into(n,abundants_up_to_reversed(n)); \\ _Antti Karttunen_, Sep 06 2018

%o (PARI) \\ see Corneth link

%o (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

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

%Y Cf. also A066874, A282568, A283550.

%K nonn

%O 0,25

%A _Reinhard Zumkeller_, Aug 25 2004

%E a(0) = 1 prepended by _David A. Corneth_, Sep 08 2018