On the representations of integers by the sextenary quadratic form x2 + y2 + z2 + 7 s2 + 7 t2 + 7 u2 and 7-cores

In this paper we derive an explicit formula for the number of representations of an integer by the sextenary form x2 + y2 + z2 + 7 s2 + 7 t2 + 7 u2. We establish the following intriguing inequalities2 ω (n + 2) ≥ a7 (n) ≥ ω (n + 2) for n ≠ 0, 2, 6, 16 . Here a7 (n) is the number of partitions of n that are 7-cores and ω (n) is the number of representations of n by the sextenary form (x2 + y2 + z2 + 7 s2 + 7 t2 + 7 u2) / 8 with x, y, z, s, t and u being odd positive integers. © 2008 Elsevier Inc. All rights reserved.