%I #5 Jun 02 2019 00:50:04

%S 1,1,2,3,5,7,10,14,19,26,34,46,58,77,93,122,146,188,217,282,327,410,

%T 470,596,673,848,947,1178,1325,1629,1798,2213,2444,2962,3247,3935,

%U 4292,5149,5579,6674,7247,8590,9221,10964,11804,13870,14843,17480,18675,21866

%N Number of integer partitions of n such that every set of distinct parts has a different sum.

%C A knapsack partition (A108917, A299702) is an integer partition such that every submultiset has a different sum. The one non-knapsack partition counted under a(4) is (2,1,1).

%e The a(1) = 1 through a(7) = 14 partitions:

%e (1) (2) (3) (4) (5) (6) (7)

%e (11) (21) (22) (32) (33) (43)

%e (111) (31) (41) (42) (52)

%e (211) (221) (51) (61)

%e (1111) (311) (222) (322)

%e (2111) (411) (331)

%e (11111) (2211) (421)

%e (3111) (511)

%e (21111) (2221)

%e (111111) (4111)

%e (22111)

%e (31111)

%e (211111)

%e (1111111)

%e The three non-knapsack partitions counted under a(6) are:

%e (2,2,1,1)

%e (3,1,1,1)

%e (2,1,1,1,1)

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@Plus@@@Subsets[Union[#]]&]],{n,0,20}]

%Y Dominates A108917.

%Y Cf. A002033, A034444, A196723, A275972, A276024, A299702, A325592, A325856, A325863, A325864, A325865, A325877.

%K nonn

%O 0,3

%A _Gus Wiseman_, May 31 2019