%I #17 Sep 13 2023 11:48:41

%S 0,1,3,5,10,14,27,35,61,83,128,166,264,327,482,632,882,1110,1565,1938,

%T 2663,3339,4401,5471,7290,8921,11555,14291,18280,22303,28507,34507,

%U 43534,52882,65798,79621,98932,118629,146072,175562,214708,256351,312583,371779

%N Number of integer partitions with sum <= n whose distinct parts can be linearly combined using nonnegative coefficients to obtain n.

%e The partition (4,2,2) cannot be linearly combined to obtain 9, so is not counted under a(9). On the other hand, the same partition (4,2,2) has distinct parts {2,4} and has 10 = 1*2 + 2*4, so is counted under a(10).

%e The a(1) = 1 through a(5) = 14 partitions:

%e (1) (1) (1) (1) (1)

%e (2) (3) (2) (5)

%e (11) (11) (4) (11)

%e (21) (11) (21)

%e (111) (21) (31)

%e (22) (32)

%e (31) (41)

%e (111) (111)

%e (211) (211)

%e (1111) (221)

%e (311)

%e (1111)

%e (2111)

%e (11111)

%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];

%t Table[Length[Select[Join@@Array[IntegerPartitions,n],combs[n,Union[#]]!={}&]],{n,0,10}]

%o (Python)

%o from sympy.utilities.iterables import partitions

%o def A365379(n):

%o a = {tuple(sorted(set(p))) for p in partitions(n)}

%o return sum(1 for m in range(1,n+1) for b in partitions(m) if any(set(d).issubset(set(b)) for d in a)) # _Chai Wah Wu_, Sep 13 2023

%Y For subsets with positive coefficients we have A088314, complement A088528.

%Y The case of strict partitions with positive coefficients is also A088314.

%Y The version for subsets is A365073, complement A365380.

%Y The case of strict partitions is A365311, complement A365312.

%Y The complement is counted by A365378.

%Y A000041 counts integer partitions, strict A000009.

%Y A008284 counts partitions by length, strict A008289.

%Y A116861 and A364916 count linear combinations of strict partitions.

%Y A364350 counts combination-free strict partitions, non-strict A364915.

%Y A364839 counts combination-full strict partitions, non-strict A364913.

%Y Cf. A237668, A363225, A364272, A364345, A364914, A365320, A365382.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 04 2023

%E a(21)-a(43) from _Chai Wah Wu_, Sep 13 2023