

A365315


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


9



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, 65, 79, 74, 85, 84, 93, 93, 107, 100, 120, 104, 126, 121, 142, 129, 145, 140, 160, 150, 173, 154, 189, 170, 196, 176, 208, 193, 223, 202, 238, 203, 241, 227, 267, 235
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OFFSET

0,5


COMMENTS

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.


LINKS



EXAMPLE

We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
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).
The a(3) = 1 through a(10) = 15 pairs:
(1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
(1,3) (1,3) (1,3) (1,3) (1,3) (1,3) (1,3)
(1,4) (1,4) (1,4) (1,4) (1,4) (1,4)
(2,3) (1,5) (1,5) (1,5) (1,5) (1,5)
(2,4) (1,6) (1,6) (1,6) (1,6)
(2,3) (1,7) (1,7) (1,7)
(2,5) (2,3) (1,8) (1,8)
(3,4) (2,4) (2,3) (1,9)
(2,6) (2,5) (2,3)
(3,5) (2,7) (2,4)
(3,6) (2,6)
(4,5) (2,8)
(3,4)
(3,7)
(4,6)


MATHEMATICA

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}]


PROG

(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(nj):
if d>=i:
break
a.add((d, i))


CROSSREFS

For all subsets instead of just pairs we have A088314, complement A365322.
The case of nonnegative coefficients is A365314, for all subsets A365073.
A004526 counts partitions of length 2, shift right for strict.
A364350 counts combinationfree strict partitions.
Cf. A070880, A088809, A326020, A364534, A365043, A365311, A365312, A365378, A365379, A365380, A365383.


KEYWORD

nonn


AUTHOR



STATUS

approved



