OFFSET
0,6
COMMENTS
Are there only two cases of nonzero adjacent equal parts, at positions n = 9, 15?
EXAMPLE
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).
The a(5) = 2 through a(10) = 12 pairs:
(2,4) (4,5) (2,4) (3,6) (2,4) (3,6)
(3,4) (2,6) (3,7) (2,6) (3,8)
(3,5) (5,6) (2,8) (3,9)
(3,6) (5,7) (4,6) (4,7)
(4,5) (6,7) (4,7) (4,8)
(4,6) (4,8) (4,9)
(5,6) (5,6) (6,7)
(5,7) (6,8)
(5,8) (6,9)
(6,7) (7,8)
(6,8) (7,9)
(7,8) (8,9)
MATHEMATICA
combs[n_, y_]:=With[{s=Table[{k, i}, {k, y}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[Subsets[Range[n], {2}], combs[n, #]=={}&]], {n, 0, 30}]
PROG
(Python)
from itertools import count
from sympy import divisors
def A365320(n):
a = set()
for i in range(1, n+1):
if not n%i:
a.update(tuple(sorted((i, j))) for j in range(1, n+1) if j!=i)
else:
for j in count(0, i):
if j > n:
break
k = n-j
for d in divisors(k):
if d>=i:
break
a.add((d, i))
return (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 13 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 06 2023
STATUS
approved