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A365382
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Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.
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5
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 4, 2, 4, 12, 8, 20, 11, 14, 26, 43, 19, 38, 53, 51, 48, 101, 48, 124, 96, 121, 159, 134, 103, 241, 261, 244, 175, 401, 229, 488, 358, 328
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OFFSET
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0,12
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LINKS
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EXAMPLE
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The a(11) = 2 through a(18) = 8 partitions:
(5,4) . (6,5) (6,5) (7,6) (7,5) (7,4) (7,5)
(7,3) (7,4) (8,5) (9,4) (7,6) (7,6) (8,7)
(7,5) (9,4) (9,5) (8,5) (10,7)
(8,3) (10,3) (11,3) (8,7) (11,4)
(9,5) (11,5)
(9,7) (12,5)
(10,3) (13,4)
(11,4) (7,5,5)
(11,5)
(13,3)
(7,4,4)
(10,3,3)
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MATHEMATICA
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combsu[n_, y_]:=With[{s=Table[{k, i}, {k, Union[y]}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Join@@IntegerPartitions/@Range[n-1], GCD@@#==1&&combsu[n, #]=={}&]], {n, 0, 20}]
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PROG
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(Python)
from math import gcd
from sympy.utilities.iterables import partitions
a = {tuple(sorted(set(p))) for p in partitions(n)}
return sum(1 for m in range(1, n) for b in partitions(m) if gcd(*b.keys()) == 1 and not any(set(d).issubset(set(b)) for d in a)) # Chai Wah Wu, Sep 13 2023
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CROSSREFS
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This is the relatively prime case of A365378.
A364350 counts combination-free strict partitions, non-strict A364915.
A364839 counts combination-full strict partitions, non-strict A364913.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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