A365314 - OEIS

%I #25 Sep 13 2023 05:58:51

%S 0,0,1,3,6,8,14,14,23,24,33,28,52,36,55,58,73,53,95,62,110,94,105,81,

%T 165,105,133,132,176,112,225,123,210,174,192,186,306,157,223,218,328,

%U 180,354,192,324,315,288,216,474,260,383,311,404,254,491,338,511,360

%N Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.

%C Is there only one case of nonzero adjacent equal parts, at position n = 6?

%H Chai Wah Wu, <a href="/A365314/b365314.txt">Table of n, a(n) for n = 0..10000</a>

%e We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).

%e The a(2) = 1 through a(7) = 14 pairs:

%e   (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)

%e          (1,3)  (1,3)  (1,3)  (1,3)  (1,3)

%e          (2,3)  (1,4)  (1,4)  (1,4)  (1,4)

%e                 (2,3)  (1,5)  (1,5)  (1,5)

%e                 (2,4)  (2,3)  (1,6)  (1,6)

%e                 (3,4)  (2,5)  (2,3)  (1,7)

%e                        (3,5)  (2,4)  (2,3)

%e                        (4,5)  (2,5)  (2,5)

%e                               (2,6)  (2,7)

%e                               (3,4)  (3,4)

%e                               (3,5)  (3,7)

%e                               (3,6)  (4,7)

%e                               (4,6)  (5,7)

%e                               (5,6)  (6,7)

%t combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];

%t Table[Length[Select[Subsets[Range[n],{2}], combs[n,#]!={}&]],{n,0,30}]

%o (Python)

%o from itertools import count

%o from sympy import divisors

%o def A365314(n):

%o     a = set()

%o     for i in range(1,n+1):

%o         if not n%i:

%o             a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)

%o         else:

%o             for j in count(0,i):

%o                 if j > n:

%o                     break

%o                 k = n-j

%o                 for d in divisors(k):

%o                     if d>=i:

%o                         break

%o                     a.add((d,i))

%o     return len(a) # _Chai Wah Wu_, Sep 12 2023

%Y The unrestricted version is A000217, ranks A001358.

%Y For all subsets instead of just pairs we have A365073, complement A365380.

%Y For strict partitions we have A365311, complement A365312.

%Y The case of positive coefficients is A365315, for all subsets A088314.

%Y The binary complement is A365320, positive A365321.

%Y For partitions we have A365379, complement A365378.

%Y A004526 counts partitions of length 2, shift right for strict.

%Y A007865 counts sum-free subsets, complement A093971.

%Y A179822 and A326080 count sum-closed subsets.

%Y A364350 counts combination-free strict partitions.

%Y A364914/A365046 count combination-full subsets, complement A326083/A124506.

%Y Cf. A070880, A088809, A151897, A326020, A365043, A365322, A365383.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 05 2023