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Let k be a fixed integer > 1 and define Tk(n)= =Edk/nl. Thus TI(n) is the ordinary divisor function and Tk(n) is the number of kth powers dividing n. We derive the asymptotic behaviour as n do of Pk(n) defined by 00 00 Pk(n)Xn = II (1 Xn) n-O n-I Thus Pk(n) is the number of partitions of n where we recognize Tk(m) different colours of the integer m when it occurs as a summand in a partition. The case k = 2 is of special interest since the number f(n) of semisimple rings with n elements when n = q|lq22 ... is given by f(n) = P2(11)P2(12) .... 1. Let k be a fixed integer > 1 and define Tk(n) = E 1. dk/n Thus Tl(n) is the ordinary divisor function and Tk(n) is the number of kth powers dividing n. We shall derive the asymptotic behaviour of Pk(n) defined by 00 00 (1.0) E Pk (n)x = II (1 xn)-Tk(n) n=O nn= 1 Thus pk(n) is the number of partitions of n where we recognize Tk(m) different colours of the integer m when it occurs as a summand in a partition. The case k = 2 is of special interest since the number of semisimple rings with n elements f2(n), when (1.1) n = pyPP2... is given byf2(n) = P2(PI)P2(P2) . . . [1]. Also, when k is large, we expectpk(n) to approach.p(n), the number of ordinary partitions. A generating function for Tk(n) is given by 00 (1.2) k(n)n (s) (ks). n= 1 LEMMA 1. If k > 1, N E Tk(n) =(k)N + O{Nl/k} n= Received by the editors October 24, 1974. AMS (MOS) subject classifications (1970). Primary lOJ20; Secondary 15A17, IOA45. t American Mathematical Society 1977

On certain weighted partitions and finite semisimple rings