OFFSET
0,3
EXAMPLE
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).
The a(1) = 1 through a(5) = 14 partitions:
(1) (1) (1) (1) (1)
(2) (3) (2) (5)
(11) (11) (4) (11)
(21) (11) (21)
(111) (21) (31)
(22) (32)
(31) (41)
(111) (111)
(211) (211)
(1111) (221)
(311)
(1111)
(2111)
(11111)
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@Array[IntegerPartitions, n], combs[n, Union[#]]!={}&]], {n, 0, 10}]
PROG
(Python)
from sympy.utilities.iterables import partitions
def A365379(n):
a = {tuple(sorted(set(p))) for p in partitions(n)}
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 04 2023
EXTENSIONS
a(21)-a(43) from Chai Wah Wu, Sep 13 2023
STATUS
approved