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%I #21 Sep 14 2023 01:10:23
%S 0,0,0,1,2,4,5,8,10,12,15,18,20,24,28,28,35,37,42,44,49,49,60,59,66,
%T 65,79,74,85,84,93,93,107,100,120,104,126,121,142,129,145,140,160,150,
%U 173,154,189,170,196,176,208,193,223,202,238,203,241,227,267,235
%N Number of unordered pairs of distinct positive integers <= n that can be linearly combined using positive coefficients to obtain n.
%C We consider (for example) that 2x + y + 3z is a positive linear combination of (x,y,z), but 2x + y is not, as the coefficient of z is 0.
%H Chai Wah Wu, <a href="/A365315/b365315.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 For the pair p = (2,3), we have 4 = 2*2 + 0*3, so p is counted under A365314(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is not counted under a(4).
%e The a(3) = 1 through a(10) = 15 pairs:
%e (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
%e (1,3) (1,3) (1,3) (1,3) (1,3) (1,3) (1,3)
%e (1,4) (1,4) (1,4) (1,4) (1,4) (1,4)
%e (2,3) (1,5) (1,5) (1,5) (1,5) (1,5)
%e (2,4) (1,6) (1,6) (1,6) (1,6)
%e (2,3) (1,7) (1,7) (1,7)
%e (2,5) (2,3) (1,8) (1,8)
%e (3,4) (2,4) (2,3) (1,9)
%e (2,6) (2,5) (2,3)
%e (3,5) (2,7) (2,4)
%e (3,6) (2,6)
%e (4,5) (2,8)
%e (3,4)
%e (3,7)
%e (4,6)
%t combp[n_,y_]:=With[{s=Table[{k,i},{k,y},{i,1,Floor[n/k]}]},Select[Tuples[s],Total[Times@@@#]==n&]];
%t Table[Length[Select[Subsets[Range[n],{2}],combp[n,#]!={}&]],{n,0,30}]
%o (Python)
%o from itertools import count
%o from sympy import divisors
%o def A365315(n):
%o a = set()
%o for i in range(1,n+1):
%o for j in count(i,i):
%o if j >= n:
%o break
%o for d in divisors(n-j):
%o if d>=i:
%o break
%o a.add((d,i))
%o return len(a) # _Chai Wah Wu_, Sep 13 2023
%Y The unrestricted version is A000217, ranks A001358.
%Y For all subsets instead of just pairs we have A088314, complement A365322.
%Y For strict partitions we have A088571, complement A088528.
%Y The case of nonnegative coefficients is A365314, for all subsets A365073.
%Y The (binary) complement is A365321, nonnegative A365320.
%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, A088809, A326020, A364534, A365043, A365311, A365312, A365378, A365379, A365380, A365383.
%K nonn
%O 0,5
%A _Gus Wiseman_, Sep 06 2023
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