A365314 Number of unordered pairs of distinct positive integers <= n that can be linearly combined using nonnegative coefficients to obtain n.

0, 0, 1, 3, 6, 8, 14, 14, 23, 24, 33, 28, 52, 36, 55, 58, 73, 53, 95, 62, 110, 94, 105, 81, 165, 105, 133, 132, 176, 112, 225, 123, 210, 174, 192, 186, 306, 157, 223, 218, 328, 180, 354, 192, 324, 315, 288, 216, 474, 260, 383, 311, 404, 254, 491, 338, 511, 360

Offset: 0

Author

Gus Wiseman, Sep 05 2023

Keywords

• nonn

Comments

Is there only one case of nonzero adjacent equal parts, at position n = 6?

Examples

			We have 19 = 4*3 + 1*7, so the pair (3,7) is counted under a(19).
The a(2) = 1 through a(7) = 14 pairs:
  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)  (1,2)
         (1,3)  (1,3)  (1,3)  (1,3)  (1,3)
         (2,3)  (1,4)  (1,4)  (1,4)  (1,4)
                (2,3)  (1,5)  (1,5)  (1,5)
                (2,4)  (2,3)  (1,6)  (1,6)
                (3,4)  (2,5)  (2,3)  (1,7)
                       (3,5)  (2,4)  (2,3)
                       (4,5)  (2,5)  (2,5)
                              (2,6)  (2,7)
                              (3,4)  (3,4)
                              (3,5)  (3,7)
                              (3,6)  (4,7)
                              (4,6)  (5,7)
                              (5,6)  (6,7)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..10000

Crossrefs

The unrestricted version is A000217, ranks A001358.

For all subsets instead of just pairs we have A365073, complement A365380.

For strict partitions we have A365311, complement A365312.

The case of positive coefficients is A365315, for all subsets A088314.

The binary complement is A365320, positive A365321.

For partitions we have A365379, complement A365378.

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

A007865 counts sum-free subsets, complement A093971.

A179822 and A326080 count sum-closed subsets.

A364350 counts combination-free strict partitions.

A364914/A365046 count combination-full subsets, complement A326083/A124506.

Cf. A070880, A088809, A151897, A326020, A365043, A365322, A365383.

Programs

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}]

Python

from itertools import count
from sympy import divisors
def A365314(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 len(a) # Chai Wah Wu, Sep 12 2023