On the representations of integers by the sextenary quadratic form x2+y2+z2+7s2+7t2+7u2x2+y2+z2+7s2+7t2+7u2 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+7s2+7t2+7u2x2+y2+z2+7s2+7t2+7u2. We establish the following intriguing inequalities2ω(n+2)⩾a7(n)⩾ω(n+2)forn≠0,2,6,16. Here a7(n)a7(n) is the number of partitions of n   that are 7-cores and ω(n)ω(n) is the number of representations of n   by the sextenary form (x2+y2+z2+7s2+7t2+7u2)/8(x2+y2+z2+7s2+7t2+7u2)/8 with x, y, z, s, t and u being odd positive integers.