Chap. X] SUM AND NuMBER OF DiVISORS. 313 
former case v=7 (mod 8). Also the left member of the third formula 
equals 
4jE(y-l)-3^(y-9)+5^(y-25)- . . .) 
when V is odd, provided E{0) = 1/4. If A'(n) denotes the sum of those divi- 
sors of n whose complementary divisors are odd, 
A'(7i)-2A'(n-l)+2A'(n-4)-2A'(n-9)+. . . =0 or (-l)'^-^ 
according as n is not or is a square. [Cf. Lipschitz.^^] Since A'(n) =a{n) 
for n odd, we deduce a formula involving c's and A"s. 
M. Lerch"'' proved (11) and 
if F{n) =2/(d), d ranging over the divisors of n. 
K. Th. Vahlen^^^ proved Liouville's^^ formula and Jacobi's^" result. 
A. P. Minin"^ proved that 2, 8, 9, 12, 18, 8g and 12p (where g is a prime 
>2, p a prime >3) are the only numbers such that each is divisible by the 
number of its divisors and the quotient is a prime. Minin^^° found that 
1, 3, 8, 10, 18, 24 and 30 are the only numbers N for which the number of 
divisors equals the number of integers < A^ and prime to A^. 
M. Lerch^^^ considered the number x{a, b) and sum X{a, h) of the divisors 
^b oi a, proved his^°° final formula (17) and 
c c 
a, X{m — a.n, a) = 2a{x(m — an, n)—yp{m — an, a)}, 
o=l a=l 
(18) S \l/[m-an, ^) = 2 xim-an, rn), c=\ ^^-— • 
a=o \ ^/ a=o L n J 
If 6 ranges over the divisors of n, 
i sV{(a-am, n)} =2^^^^, | sV|(a-am, n)\ =2(5, m) a), 
't a=0 n a=0 
S (am, n) =nS^-(5, (m, n)). 
a=l 
TO— 1 
He quoted (p. 8) from a letter to him from Chr. Zeller that 2 a\p{m — a, a) 
a=l 
equals the sum of the remainders obtained on dividing m by the integers 
<m. 
M. Lerch^^^ proved that 
'Zxf/im+p—crn, a) =2x(w+p— o-n, n) — 2 p \, 
i:\pim-p-pn, a)=Xx{m-p-(rn, ^) -2[^:j:YjLn+fJ' 
i^Casopis, Prag, 21, 1892, 90-95, 185-190 (in Bohemian). Cf. Jahrbuch Fortschritte Math., 
24, 1892, 186-7. 
"sjour. fur Math., 112, 1893, 29. 
i^Math. Soc. Moscow, 17, 1893, 240-253. 
i2o/6id., 17, 1894, 537-544. 
"iPrag Sitzungsberichte (Math.), 1894, No. 11. 
^Ubid., No. 32. 
