A394445 a(n) is the minimum k such that every 2-coloring of {1, ..., k} contains a monochromatic solution to x + y = n*z with x, y, z pairwise distinct.

9, 9, 15, 20, 25, 31, 49, 41, 54, 61, 77, 85, 104, 113, 135, 145, 170, 181, 209, 221, 252, 265, 299, 313, 350, 365, 405, 421, 464, 481, 527, 545, 594, 613, 665, 685, 740, 761, 819, 841, 902, 925, 989

Offset

1,1

Comments

These are "distinct-variable" 2-color Rado numbers for the equation x + y = n*z. The standard (non-distinct) 2-color Rado numbers, where variables may repeat, are n*(n+1)/2 for n >= 4 (Burr and Loo, see A100542). For n=1 this is the Schur equation x+y=z with distinct variables, giving the weak Schur number WS(2)+1 = 9 (cf. A072842). For n=2 this is x+y=2*z with distinct variables, equivalent to 3-term arithmetic progressions, giving the van der Waerden number W(3;2) = 9. The sequence is not monotone: a(7) = 49 > a(8) = 41.

References

S. A. Burr and S. P. Loo, On Rado numbers II, unpublished manuscript, c. 1992.

H. Harborth and S. Maasberg, All two-color Rado numbers for a(x+y)=bz, Discrete Math., 197/198 (1999), 397-407.

S. Jones and D. Schaal, Two-color Rado numbers for x+y+c=kz, Discrete Math., 289 (2004), 63-69.

Links

Alex Towell, Table of n, a(n) for n = 1..500

Alex Towell, Rado Numbers for x+y=kz: Computation, Closed-Form Formula, and Complete Proof, Zenodo, 2026.

Formula

For n >= 8, a(n) = n*(n+3)/2 if n is odd, a(n) = (n^2 + 2*n + 2)/2 if n is even.

Example

a(3) = 15 because every 2-coloring of {1, ..., 15} contains a monochromatic solution to x+y=3z with x, y, z distinct, but the 2-coloring {2, 5, 8, 9, 11, 12, 14} red, {1, 3, 4, 6, 7, 10, 13} blue of {1, ..., 14} avoids all monochromatic solutions.

Prog

(Python)
from pysat.solvers import Solver
def a(n, max_N=2000):
    for N in range(1, max_N):
        sols = []
        for z in range(1, N+1):
            for x in range(1, N+1):
                y = n*z - x
                if 1 <= y <= N and len({x, y, z}) == 3:
                    sols.append((x, y, z))
        with Solver(name="cd195") as s:
            for i in range(1, N+1):
                s.add_clause([2*i-1, 2*i])
                s.add_clause([-(2*i-1), -(2*i)])
            for x, y, z in sols:
                for v in [lambda i: 2*i-1, lambda i: 2*i]:
                    s.add_clause([-v(x), -v(y), -v(z)])
            if not s.solve():
                return N

Crossrefs

Cf. A100542 (non-distinct variant), A072842 (weak Schur numbers), A003071 (Schur numbers).

Cf. A000096, A001844.

Sequence in context: A393829 A144418 A393818 * A003885 A344335 A168390

Adjacent sequences: A394442 A394443 A394444 * A394446 A394447 A394448

Keyword

nonn

Author

Alex Towell, Mar 20 2026

Status

approved