New inequalities for p(n) and logp(n).

Let p(n) denote the number of partitions of n. A new infinite family of inequalities for p(n) is presented. This generalizes a result by William Chen et al. From this infinite family, another infinite family of inequalities for logp(n) is derived. As an application of the latter family one, for instance obtains that for n≥120, p(n)2>(1+π24n3/2-1n2)p(n-1)p(n+1).

The Andrews-Sellers family of partition congruences.

In 1994, James Sellers conjectured an infinite family of Ramanujan type congruences for 2-colored Frobenius partitions introduced by George E. Andrews. These congruences arise modulo powers of 5. In 2002 Dennis Eichhorn and Sellers were able to settle the conjecture for powers up to 4. In this article, we prove Sellers' conjecture for all powers of 5. In addition, we discuss why the Andrews-Sellers family is significantly different from classical congruences modulo powers of primes.