CHAP, xi] MISCELLANEOUS THEOEEMS ON DIVISIBILITY. 329 Gegenbauer10a proved that the arithmetical mean of the greatest integers contained in k times the remainders on the division of n by 1, 2, . . ., n approaches as n increases. The case k=2 is due to Dirichlet. Gegenbauer11 gave formulas involving the greatest divisor tj(ri), not divisible by a, of the integer n. In particular, he gave the mean value of the greatest divisor not divisible by an ath power. L. Gegenbauer,12 employing Merten's function p, (Ch. XIX) and jR(a) = a — |a|, gave the three general formulas where x2 ranges over the divisors >n of (r— l)n+l, (r — l)n+2,. . ., rn, while a?! ranges over all positive integers for which r+n Q r n where g is the g. c. d. of r, n. Take f(x) = 1 or 0 according as x is an sth power or not. Then the functions (i) J/W, become [^J/m] and \(x), with the value 0 if the exponent of any prime factor of x is f^O, 1 (mod s), otherwise the value ( — 1)*, where cr is the number of primes occurring in x to the power fcs+1. Thus If f(x) =0 or 1 according as x is divisible by an sth power or not, the functions (1) become Q9(m) and n(-f/x), the former being the number of integers ^ m divisible by no sth power. If f(x) = 1 or 0 according as x is prime or not, the functions (1) become the number of primes ^m and a simple function a(x) ; then the third formula shows that the mean density of the primes l°aDenkschr. Akad. Wien (Math.), 49, II, 1885, 108. "Sitzungsber. Akad. Wiss. Wien (Math.), 94, 1886, II, 714. »/Wa., 97, 1888, Ha, 420-6.