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A364915
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Number of integer partitions of n such that no distinct part can be written as a nonnegative linear combination of other distinct parts.
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21
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1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 8, 12, 10, 16, 16, 19, 21, 29, 25, 37, 35, 44, 46, 60, 55, 75, 71, 90, 90, 114, 110, 140, 138, 167, 163, 217, 201, 248, 241, 298, 303, 359, 355, 425, 422, 520, 496, 594, 603, 715, 706, 834, 826, 968, 972, 1153, 1147, 1334, 1315, 1530
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OFFSET
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0,3
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LINKS
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FORMULA
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EXAMPLE
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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).
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MATHEMATICA
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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}]
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PROG
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(Python)
from sympy.utilities.iterables import partitions
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
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CROSSREFS
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For subsets instead of partitions we have A326083, complement A364914.
A007865 counts binary sum-free sets w/ re-usable parts, complement A093971.
A364912 counts linear combinations of partitions of k.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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