A365321 Number of pairs of distinct positive integers <= n that cannot be linearly combined with positive coefficients to obtain n.

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

Offset: 0

Author

Gus Wiseman, Sep 06 2023

Keywords

• nonn

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.

Examples

			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)
		

Crossrefs

The unrestricted version is A000217, ranks A001358.

For strict partitions we have A088528, complement A088314.

The (binary) complement is A365315, nonnegative A365314.

For nonnegative coefficients we have A365320, for subsets A365380.

For all subsets instead of just pairs we have A365322, complement A088314.

A004526 counts partitions of length 2, shift right for strict.

A007865 counts sum-free subsets, complement A093971.

A179822 and A326080 count sum-closed subsets.

A326083 and A124506 count combination-free subsets.

A364350 counts combination-free strict partitions.

A364914 and A365046 count combination-full subsets.

Cf. A070880, A088571, A088809, A151897, A326020, A365043, A365073, A365311, A365312, A365378, A365380.

Programs

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}]

Python

from itertools import count
from sympy import divisors
def A365321(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 (n*(n-1)>>1)-len(a) # Chai Wah Wu, Sep 12 2023