302 HISTORY OF THE THEORY OF NUMBERS. [CHAP, x L. Gegenbauer80 considered the number AO(OL) of those divisors of a which are congruent modulo k and have a complementary divisor SE! (mod &). He proved that, if p<k, <r-i-(k-p)x] -P7" J' fl-[jR fe+i If we replace a by <r 1 and subtract, we obtain expressions for A0(far p). The above formulas give, for fc-2, p = l, and formulas of Bouniakowsky.32 The same developments show that an odd number a is a prime if a-2 , r a , n = r L2(2x+l) "t"2j L2 for z^[(a 3)/2]; likewise for a = 6fc=±=l if the same equality holds when x^[(a 5)/6], with similar tests for a = 3n 1, or 4n 1. C. Runge81 proved that r(n)/n* has the limit zero as n increases indefinitely, for every e> 0. E. Catalan82 noted that, if xnp is the number of ways of decomposing a product of n distinct primes into p factors >1, order being immaterial, E. Cesaro83 considered the number Fm (x) of integers ^x which are not divisible by mth powers, and the number Tm (x) of those divisors of x which are mth powers, evaluated sums involving these and other functions, and determined mean values and probabilities relating to the greatest square divisor of an arbitrary integer. R. Lipschitz84 considered the sum k(m) of the odd divisors of m increased by half the sum of the even divisors, and the function l(m) obtained by interchanging the words "even," "odd." He proved that k(m)-2k(m-l)+2k(m-9)-~. . . =(-l)m-lm or 0, according as m is a square or is not; Z(m)+Z(m-l)+Z(7n~3)+Z(m-6)+ . . . = -m or 0, according as m is a triangular number or is not; K(m) = t(l) +*(2) + . . . +k(m) = [m] + g] +3 [|] +2 g] + . . . +/tg] , L(m) =Z 80Sitzungsberichte Ak. Wien. (Math.), 91, II, 1885, 1194-1201. 81Acta Math., 7, 1885, 181-3. »2M6m. soc. roy. sc. Li^ge, (2), 12, 1885, 18-20; Melanges Math., 1868, 18. 83Annali di Mat., (2), 13, 1885, 251-268. Reprint "Excursions arith. a 1'infini," 17-34. "Comptes Rendus Paris, 100, 1885, 845. Cf. Glaisher118, also Fergola21 of Ch. XI, Vol. II.