302 History of the Theory of Numbers. [Chap, x 
L. Gegenbauer^" considered the number Ao{a) of those di\'isors of a 
which are congruent modulo k and have a complementary divisor =1 
(mod k). He proved that, if p<k, 
If we replace (t by <t — 1 and subtract, we obtain expressions for Ao{k(T—p). 
The above formulas give, for k = 2, p = l, 
and formulas of Bouniakowsky.^^ The same developments show that an 
odd number a is a prime if 
L2(2a:+1) ^2j L2(2x+l)^2j 
for x^[(a — Z)/2]; likewise for a = 6fc=*=l if the same equality holds when 
x^[(a — 5)/Q], with similar tests for a = 3n — 1, or 4n — 1. 
C. Runge^^ proved that T{n)/n* has the limit zero as n increases indefi- 
nitely, for every e>0. 
E. Catalan^- noted that, if x^p is the number of ways of decomposing a 
product of n distinct primes into p factors >1, order being immaterial, 
x„p = px„_ip+x„_i,_i = jp'-^-(PTi)(p-ir-^+(^-^)(p-2r-^-...±i} 
-^{(p-l!). 
E. Cesaro^ considered the number F„ (x) of integers ^x which are not 
divisible by mth powers, and the number T^ (x) of those di\'isors 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. Lipschitz^ considered the sum k{m) of the odd di\'isors of m increased 
by half the sum of the even di\'isors, and the function l(m) obtained by 
interchanging the words "even," "odd." He proved that 
k{m)-2k{m-l)+2k{m-9)- . . . =(-1)"-'^ or 0, 
according as m is a square or is not; 
l{m)+l{m-l)+l{77i-S)+l{m-6)+. . . = -m or 0, 
according as m is a triangular number or is not ; 
XW=A'(1)+A'(2)+...+A-W = H+[|]+3[|]+2[^] + ...+m[^], 
L(m) = /(l)+Z(2)+ . . . +l{m) = -[m]-^2[f\ -^[f\+^[f\ -■■■> 
"Sitzunpsberichte Ak. Wien. (Math.), 91, II, 1885, 1194-1201. "Acta Math., 7, 1885, 181-3. 
"M^m. 80C. roy. sc. Lifege, (2), 12, 1885, 18-20; Melanges Math., 1868, 18. 
"Annali di Mat., (2), 13, 1885, 251-268. Reprint "Excursions arith. k Tinfini," 17-34. 
"Comptes Rendus Paris, 100, 1885, 845. Cf. Glaisher"«, also Fergola" of Ch. XI, Vol. II. 
