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A365320 Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.

%I A365320 #14 Sep 14 2023 01:11:14
%S A365320 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 A365320 111,195,192,219,202,294,210,342,286,354,369,409,324,509,480,523,452,
%U A365320 640,507,711,622,675,747,865,654,916,842,964,922,1124,940,1147,1029
%N A365320 Number of pairs of distinct positive integers <= n that cannot be linearly combined with nonnegative coefficients to obtain n.
%C A365320 Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?
%e A365320 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 A365320 The a(5) = 2 through a(10) = 12 pairs:
%e A365320   (2,4)  (4,5)  (2,4)  (3,6)  (2,4)  (3,6)
%e A365320   (3,4)         (2,6)  (3,7)  (2,6)  (3,8)
%e A365320                 (3,5)  (5,6)  (2,8)  (3,9)
%e A365320                 (3,6)  (5,7)  (4,6)  (4,7)
%e A365320                 (4,5)  (6,7)  (4,7)  (4,8)
%e A365320                 (4,6)         (4,8)  (4,9)
%e A365320                 (5,6)         (5,6)  (6,7)
%e A365320                               (5,7)  (6,8)
%e A365320                               (5,8)  (6,9)
%e A365320                               (6,7)  (7,8)
%e A365320                               (6,8)  (7,9)
%e A365320                               (7,8)  (8,9)
%t A365320 combs[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,0,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t A365320 Table[Length[Select[Subsets[Range[n],{2}],combs[n,#]=={}&]],{n,0,30}]
%o A365320 (Python)
%o A365320 from itertools import count
%o A365320 from sympy import divisors
%o A365320 def A365320(n):
%o A365320     a = set()
%o A365320     for i in range(1,n+1):
%o A365320         if not n%i:
%o A365320             a.update(tuple(sorted((i,j))) for j in range(1,n+1) if j!=i)
%o A365320         else:
%o A365320             for j in count(0,i):
%o A365320                 if j > n:
%o A365320                     break
%o A365320                 k = n-j
%o A365320                 for d in divisors(k):
%o A365320                     if d>=i:
%o A365320                         break
%o A365320                     a.add((d,i))
%o A365320     return (n*(n-1)>>1)-len(a) # _Chai Wah Wu_, Sep 13 2023
%Y A365320 The unrestricted version is A000217, ranks A001358.
%Y A365320 For strict partitions we have A365312, complement A365311.
%Y A365320 The (binary) complement is A365314, positive A365315.
%Y A365320 The case of positive coefficients is A365321, for all subsets A365322.
%Y A365320 For partitions we have A365378, complement A365379.
%Y A365320 For all subsets instead of just pairs we have A365380, complement A365073.
%Y A365320 A004526 counts partitions of length 2, shift right for strict.
%Y A365320 A007865 counts sum-free subsets, complement A093971.
%Y A365320 A179822 and A326080 count sum-closed subsets.
%Y A365320 A326083 and A124506 appear to count combination-free subsets.
%Y A365320 A364350 counts combination-free strict partitions.
%Y A365320 A364914 and A365046 count combination-full subsets.
%Y A365320 Cf. A070880, A088314, A088809, A151897, A326020, A364839.
%K A365320 nonn
%O A365320 0,6
%A A365320 _Gus Wiseman_, Sep 06 2023