A394178 Triangle read by rows: T(n, k) is the Frobenius number for n and k if n and k coprime, and 0 otherwise, for 1 <= k <= n.

-1, -1, 0, -1, 1, 0, -1, 0, 5, 0, -1, 3, 7, 11, 0, -1, 0, 0, 0, 19, 0, -1, 5, 11, 17, 23, 29, 0, -1, 0, 13, 0, 27, 0, 41, 0, -1, 7, 0, 23, 31, 0, 47, 55, 0, -1, 0, 17, 0, 0, 0, 53, 0, 71, 0, -1, 9, 19, 29, 39, 49, 59, 69, 79, 89, 0, -1, 0, 0, 0, 43, 0, 65, 0, 0, 0, 109, 0

Offset

1,9

Comments

By Sylvester's two-coin formula, for gcd(n, k) = 1 the Frobenius number is g(n, k) = n*k - n - k. For gcd(n, k) > 1 the Frobenius number is undefined and we record 0.

All nonzero terms are odd, every row n > 2 contains an even number of nonzero terms.

Links

Felix Huber, Rows n=1..141, flattened

Formula

G.f.: Sum_{n>=1} T(n + m, n)*x^n = Sum_{d|m} mu(d)*[d^2*y*(1 + y)/(1 - y)^3 + d*(m - 2)*y/(1 - y)^2 - (m - 1)*y/(1 - y) - y/(1 - y)], where y = x^d.

T(n, k) = (A079904(n - 1, k - 1) - 1)*A054521(n, k).

T(n, k) = (A079904(n - 1, k - 1) - 1)*Sum_{d | gcd(n, k)} A008683(d).

T(n + 1, n) = (A002378(n - 1) - 1)*A054521(n, m).

Sum_{k=1..n} T(n, k) = A000010(n)*A000217(n - 2) - A000010(n) for n > 1.

T(n,k) = A380697(2^(n-2)+2^(k-2)) if 2 <= k <= n. - Pontus von Brömssen, May 06 2026

Example

The triangle begins:
  n\k |  1   2   3   4   5   6   7   8   9  10  11  12
  ----+------------------------------------------------
   1  | -1
   2  | -1   0
   3  | -1   1   0
   4  | -1   0   5   0
   5  | -1   3   7  11   0
   6  | -1   0   0   0  19   0
   7  | -1   5  11  17  23  29   0
   8  | -1   0  13   0  27   0  41   0
   9  | -1   7   0  23  31   0  47  55   0
  10  | -1   0  17   0   0   0  53   0  71   0
  11  | -1   9  19  29  39  49  59  69  79  89   0
  12  | -1   0   0   0  43   0  65   0   0   0 109   0

Maple

T := (n, k) -> `if`(igcd(n, k) <> 1, 0, n*k - n - k):
seq(seq(T(n, k), k = 1 .. n), n = 1 .. 12);

Mathematica

A394178[n_, k_] := If[CoprimeQ[n, k], n*k - n - k, 0];
Table[A394178[n, k], {n, 15}, {k, n}] (* Paolo Xausa, May 12 2026 *)

Crossrefs

Cf. A000010, A000217, A002378, A008683, A023896, A037165, A054521, A079904.

Cf. A069755, A124506, A199711, A293176, A319608, A380697, A395177.

Sequence in context: A019729 A261851 A117015 * A325736 A370012 A370013

Adjacent sequences: A394175 A394176 A394177 * A394179 A394180 A394181

Keyword

tabl,sign,easy

Author

Felix Huber, May 05 2026

Status

approved