On Higher Power Moment Of The Number On Non-isomorphic Abelian Groups

Let the arithmetical function a(n) denote the number of non-isomorphic a-belian groups with n elements. It is well known that a(n) is a positive, integer-valued multiplication function, with the property that a(pα)=P (α), for everyprime p and every integer α≥1. P (α) is the number of unrestricted partitions ofα, especially we havea(1)=1, a(p)=1, a(p2)=2, a(p3)=3,a(p4)=5, a(p5)=7, a(p6)=11, a(p7)=15.(1)Many experts have studied the mean value of a(n) thoroughly:P.Erdo¨s and G.Szekeres[1]first proved thatKendall and Rankin[2]proved that where HP (j)1(t) is P (j)1power polynomial of t.Aleksandar Ivic[12]proved thatIn this paper, we shall investigate the mean value of a2(n) in square-full num-ber set and in cube-full number set, namely:Suppose k≥2is a fixed positive integer. The standard decomposition of1...ps, n is called square-full number when αj≥2(j=1,...s) andcube-full number when αj≥2(j=1,...s). Here f2(n) expresses the characteristicfunction of square-full number set and f3(n) expresses the characteristic function of cube-full number set.We shall prove the following:Theorem1We have the following asymptotic formulawhere§Pj(t)(t=1,2) is j power polynomial of t and σ0=29353Theorem2We have the following asymptotic formulaIn addition, we also investigate the mean value of1/a2(n),we have the followingtheorem:Theorem3Suppose N≥1are fixed integers,thenwhere d0, d1,..., dNare computable constants.In the end of the paper, we prove a short interval result for the function a2(n),as the following:...