cp's OEIS Frontend

A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.

%I A365382 #17 Sep 13 2023 14:16:53
%S A365382 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,
%T A365382 101,48,124,96,121,159,134,103,241,261,244,175,401,229,488,358,328
%N A365382 Number of relatively prime integer partitions with sum < n that cannot be linearly combined using nonnegative coefficients to obtain n.
%e A365382 The a(11) = 2 through a(18) = 8 partitions:
%e A365382   (5,4)  .  (6,5)  (6,5)   (7,6)  (7,5)   (7,4)     (7,5)
%e A365382   (7,3)     (7,4)  (8,5)   (9,4)  (7,6)   (7,6)     (8,7)
%e A365382             (7,5)  (9,4)          (9,5)   (8,5)     (10,7)
%e A365382             (8,3)  (10,3)         (11,3)  (8,7)     (11,4)
%e A365382                                           (9,5)     (11,5)
%e A365382                                           (9,7)     (12,5)
%e A365382                                           (10,3)    (13,4)
%e A365382                                           (11,4)    (7,5,5)
%e A365382                                           (11,5)
%e A365382                                           (13,3)
%e A365382                                           (7,4,4)
%e A365382                                           (10,3,3)
%t A365382 combsu[n_,y_]:=With[{s=Table[{k,i},{k,Union[y]},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365382 Table[Length[Select[Join@@IntegerPartitions/@Range[n-1],GCD@@#==1&&combsu[n,#]=={}&]],{n,0,20}]
%o A365382 (Python)
%o A365382 from math import gcd
%o A365382 from sympy.utilities.iterables import partitions
%o A365382 def A365382(n):
%o A365382     a = {tuple(sorted(set(p))) for p in partitions(n)}
%o A365382     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
%Y A365382 Relatively prime partitions are counted by A000837, ranks A289509.
%Y A365382 This is the relatively prime case of A365378.
%Y A365382 A000041 counts integer partitions, strict A000009.
%Y A365382 A008284 counts partitions by length, strict A008289.
%Y A365382 A116861 and A364916 count linear combinations of strict partitions.
%Y A365382 A364350 counts combination-free strict partitions, non-strict A364915.
%Y A365382 A364839 counts combination-full strict partitions, non-strict A364913.
%Y A365382 Cf. A007359, A289508, A364345, A365073, A365312, A365379, A365380, A365383.
%K A365382 nonn,more
%O A365382 0,12
%A A365382 _Gus Wiseman_, Sep 08 2023
%E A365382 a(21)-a(45) from _Chai Wah Wu_, Sep 13 2023