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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
OFFSET
0,12
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.
Sequence in context: A004174 A348874 A300328 * A200291 A049797 A116578
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Sep 08 2023
EXTENSIONS
a(21)-a(45) from Chai Wah Wu, Sep 13 2023
STATUS
approved