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
Chai Wah Wu, Table of n, a(n) for n = 0..10000
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
def A365315(n):
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))
return len(a) # Chai Wah Wu, Sep 13 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 06 2023
STATUS
approved