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A365321
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Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.
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10
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0, 0, 1, 2, 4, 6, 10, 13, 18, 24, 30, 37, 46, 54, 63, 77, 85, 99, 111, 127, 141, 161, 171, 194, 210, 235, 246, 277, 293, 322, 342, 372, 389, 428, 441, 491, 504, 545, 561, 612, 635, 680, 701, 753, 773, 836, 846, 911, 932, 1000, 1017, 1082, 1103, 1176, 1193
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OFFSET
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0,4
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COMMENTS
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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.
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LINKS
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EXAMPLE
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For the pair p = (2,3) we have 4 = 2*2 + 0*3, so p is not counted under A365320(4), but it is not possible to write 4 as a positive linear combination of 2 and 3, so p is counted under a(4).
The a(2) = 1 through a(7) = 13 pairs:
(1,2) (1,3) (1,4) (1,5) (1,6) (1,7)
(2,3) (2,3) (2,4) (2,3) (2,4)
(2,4) (2,5) (2,5) (2,6)
(3,4) (3,4) (2,6) (2,7)
(3,5) (3,4) (3,5)
(4,5) (3,5) (3,6)
(3,6) (3,7)
(4,5) (4,5)
(4,6) (4,6)
(5,6) (4,7)
(5,6)
(5,7)
(6,7)
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MATHEMATICA
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combp[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 1, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n], {2}], combp[n, #]=={}&]], {n, 0, 30}]
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PROG
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(Python)
from itertools import count
from sympy import divisors
a = set()
for i in range(1, n+1):
for j in count(i, i):
if j >= n:
break
for d in divisors(n-j):
if d>=i:
break
a.add((d, i))
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CROSSREFS
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For all subsets instead of just pairs we have A365322, complement A088314.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combination-free strict partitions.
Cf. A070880, A088571, A088809, A151897, A326020, A365043, A365073, A365311, A365312, A365378, A365380.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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