Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #13 Sep 26 2023 14:58:01
%S 1,2,5,10,18,32,51,82,122,188,262,392,529,750,997,1404,1784,2452,3123,
%T 4164,5239,6916,8499,11112,13693,17482,21257,27162,32581,41114,49606,
%U 61418,73474,91086,107780,132874,157359,191026,225159,274110,320691,386722,453875
%N Number of special sums of integer partitions of n.
%C A special sum of an integer partition y is a number n >= 0 such that exactly one submultiset of y sums to n.
%F a(n) = A301854(n) + A000041(n).
%e The a(4) = 18 special positive subset-sums:
%e 0<=(4), 4<=(4),
%e 0<=(22), 2<=(22), 4<=(22),
%e 0<=(31), 1<=(31), 3<=(31), 4<=(31),
%e 0<=(211), 1<=(211), 3<=(211), 4<=(211),
%e 0<=(1111), 1<=(1111), 2<=(1111), 3<=(1111), 4<=(1111).
%t uqsubs[y_]:=Join@@Select[GatherBy[Union[Subsets[y]],Total],Length[#]===1&];
%t Table[Total[Length/@uqsubs/@IntegerPartitions[n]],{n,25}]
%Y Cf. A000712, A108917, A122768, A275972, A276024, A284640, A299701, A299702, A299729, A301829, A301830, A301854.
%K nonn
%O 0,2
%A _Gus Wiseman_, May 18 2018
%E More terms from _Alois P. Heinz_, May 18 2018
%E a(36)-a(42) from _Chai Wah Wu_, Sep 26 2023