%I #14 Sep 28 2023 04:41:17
%S 0,0,1,1,3,3,6,7,12,14,22,25,38,46,64,76,106,124,167,199,261,309,402,
%T 471,604,714,898,1053,1323,1542,1911,2237,2745,3201,3913,4536,5506,
%U 6402,7706,8918,10719,12364,14760,17045,20234,23296,27600,31678,37365,42910,50371,57695,67628,77300,90242,103131,119997
%N Number of incomplete integer partitions of n, meaning not every number from 0 to n is the sum of some submultiset.
%C The complement (complete partitions) is A126796.
%H Joerg Arndt, <a href="/A365924/b365924.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = A000041(n) - A126796(n).
%e The a(0) = 0 through a(8) = 12 partitions:
%e . . (2) (3) (4) (5) (6) (7) (8)
%e (2,2) (3,2) (3,3) (4,3) (4,4)
%e (3,1) (4,1) (4,2) (5,2) (5,3)
%e (5,1) (6,1) (6,2)
%e (2,2,2) (3,2,2) (7,1)
%e (4,1,1) (3,3,1) (3,3,2)
%e (5,1,1) (4,2,2)
%e (4,3,1)
%e (5,2,1)
%e (6,1,1)
%e (2,2,2,2)
%e (5,1,1,1)
%t nmz[y_]:=Complement[Range[Total[y]],Total/@Subsets[y]];
%t Table[Length[Select[IntegerPartitions[n],Length[nmz[#]]>0&]],{n,0,15}]
%Y For parts instead of sums we have A047967/A365919, ranks A080259/A055932.
%Y The complement is A126796, ranks A325781, strict A188431.
%Y These partitions have ranks A365830.
%Y The strict case is A365831.
%Y Row sums of A365923 without the first column, strict A365545.
%Y A000041 counts integer partitions, strict A000009.
%Y A046663 counts partitions w/o a submultiset summing to k, strict A365663.
%Y A276024 counts positive subset-sums of partitions, strict A284640.
%Y A325799 counts non-subset-sums of prime indices.
%Y A364350 counts combination-free strict partitions.
%Y A365543 counts partitions with a submultiset summing to k, strict A365661.
%Y Cf. A002865, A006827, A018818, A264401, A299701, A304792, A364272, A365658, A365918, A365921.
%K nonn
%O 0,5
%A _Gus Wiseman_, Sep 26 2023