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 A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n. 5
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 LINKS Table of n, a(n) for n=0..45. EXAMPLE 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) MATHEMATICA 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}] PROG (Python) from math import gcd from sympy.utilities.iterables import partitions def A365382(n): 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 CROSSREFS Relatively prime partitions are counted by A000837, ranks A289509. This is the relatively prime case of A365378. A000041 counts integer partitions, strict A000009. A008284 counts partitions by length, strict A008289. A116861 and A364916 count linear combinations of strict partitions. A364350 counts combination-free strict partitions, non-strict A364915. A364839 counts combination-full strict partitions, non-strict A364913. Cf. A007359, A289508, A364345, A365073, A365312, A365379, A365380, A365383. Sequence in context: A004174 A348874 A300328 * A200291 A049797 A116578 Adjacent sequences: A365379 A365380 A365381 * A365383 A365384 A365385 KEYWORD nonn,more AUTHOR Gus Wiseman, Sep 08 2023 EXTENSIONS a(21)-a(45) from Chai Wah Wu, Sep 13 2023 STATUS approved

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Last modified August 4 22:06 EDT 2024. Contains 374934 sequences. (Running on oeis4.)