A365320 - OEIS

%I #14 Sep 14 2023 01:11:14

%S 0,0,0,0,0,2,1,7,5,12,12,27,14,42,36,47,47,83,58,109,80,116,126,172,

%T 111,195,192,219,202,294,210,342,286,354,369,409,324,509,480,523,452,

%U 640,507,711,622,675,747,865,654,916,842,964,922,1124,940,1147,1029

%N Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.

%C Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?

%e The pair p = (3,6) cannot be linearly combined to obtain 8 or 10, so p is counted under a(8) and a(10), but we have 9 = 1*3 + 1*6 or 9 = 3*3 + 0*6, so p not counted under a(9).

%e The a(5) = 2 through a(10) = 12 pairs:

%e   (2,4)  (4,5)  (2,4)  (3,6)  (2,4)  (3,6)

%e   (3,4)         (2,6)  (3,7)  (2,6)  (3,8)

%e                 (3,5)  (5,6)  (2,8)  (3,9)

%e                 (3,6)  (5,7)  (4,6)  (4,7)

%e                 (4,5)  (6,7)  (4,7)  (4,8)

%e                 (4,6)         (4,8)  (4,9)

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

%e                               (5,7)  (6,8)

%e                               (5,8)  (6,9)

%e                               (6,7)  (7,8)

%e                               (6,8)  (7,9)

%e                               (7,8)  (8,9)

%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 A365320(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 (n*(n-1)>>1)-len(a) # _Chai Wah Wu_, Sep 13 2023

%Y The unrestricted version is A000217, ranks A001358.

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

%Y The (binary) complement is A365314, positive A365315.

%Y The case of positive coefficients is A365321, for all subsets A365322.

%Y For partitions we have A365378, complement A365379.

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

%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 A326083 and A124506 appear to count combination-free subsets.

%Y A364350 counts combination-free strict partitions.

%Y A364914 and A365046 count combination-full subsets.

%Y Cf. A070880, A088314, A088809, A151897, A326020, A364839.

%K nonn

%O 0,6

%A _Gus Wiseman_, Sep 06 2023