OFFSET
0,3
EXAMPLE
The a(1) = 1 through a(10) = 8 partitions (A=10):
1 2 3 4 5 6 7 8 9 A
11 111 22 32 33 43 44 54 55
1111 11111 222 52 53 72 64
111111 322 332 333 73
1111111 2222 522 433
11111111 3222 3322
111111111 22222
1111111111
The partition (5,4,3) has no part that can be written as a nonnegative linear combination of the others, so is counted under a(12).
The partition (6,4,3,2) has 6=4+2, or 6=3+3, or 6=2+2+2, or 4=2+2, so is not counted under a(15).
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[IntegerPartitions[n], Function[ptn, !Or@@Table[combs[ptn[[k]], Delete[ptn, k]]!={}, {k, Length[ptn]}]]@*Union]], {n, 0, 15}]
PROG
(Python)
from sympy.utilities.iterables import partitions
def A364915(n):
if n <= 1: return 1
alist, c = [set(tuple(sorted(set(p))) for p in partitions(i)) for i in range(n)], 1
for p in partitions(n, k=n-1):
s = set(p)
if not any(set(t).issubset(s-{q}) for q in s for t in alist[q]):
c += 1
return c # Chai Wah Wu, Sep 23 2023
KEYWORD
nonn
AUTHOR
Gus Wiseman, Aug 22 2023
EXTENSIONS
a(37)-a(59) from Chai Wah Wu, Sep 25 2023
STATUS
approved