Let a1,a2,…,ak be positive integers with gcd(a1,a2,…,ak)=1. Frobenius number is the largest integer that NOT representable in terms of a1,a2,…,ak. When k?3, there no explicit formula general, but some formulae may exist for special sequences a1,a2,…,ak, including, those forming arithmetic progressions and their modifications. In this paper, we give power weighted sum nonrepresentable integers. ...

In this note we show a simple formula for the coefficients of polynomial associated with sums powers terms an arbitrary arithmetic progression. This consists double sum involving only ordinary binomial and powers. Arguably, is simplest that can probably be found said coefficients. Furthermore, give explicit Bernoulli polynomials Stirling numbers first second kind.

Enumerating formulae are constructed which count the number of partitions of a positive integer into positive summands in arithmetic progression with common difference D. These enumerating formulae (denoted pD(n)) which are given in terms of elementary divisor functions together with auxiliary arithmetic functions (to be defined) are then used to establish a known characterisation for an intege...

Let G be a properly colored bipartite graph. A rainbow matching of G is such a matching in which no two edges have the same color. Let G be a properly colored bipartite graph with bipartition ( X , Y ) and . We show that if   = G k   3   7 max , 4 k X Y  , then G has a rainbow coloring of size at least 3 4 k       .

Several renowned open conjectures in combinatorics and number theory involve arithmetic progressions. Van der Waerden famously proved in 1927 that for each positive integer k there exists a least positive integer w(k) such that any 2-coloring of 1, . . . , w(k) produces a monochromatic k-term arithmetic progression. The best known upper bound for w(k) is due to Gowers and is quite large. Ron Gr...