CHAP.X] SUM AND NlJMBEK OF DlVISORS. 313 former case v=7 (mod 8). Also the left member of the third formula equals when v is odd, provided E(Q) = 1/4. If A'(n) denotes the sum of those divisors of n whose complementary divisors are odd, A'(n)-2A'(n~l)+2A\'n-4)-2A'(n-9)-|-. . . =0 or (-I)*"1*, according as n is not or is a square. [CL Lipschitz.84] Since A'(n) = <j(n) for ?i odd, we deduce a formula involving <r's and A"s. M. Lerch117 proved (11) and if /P(n) =S/(d), d ranging over the divisors of n. K. Th. Vahlen118 proved Liouville's23 formula and Jacobi's10 result. A. P. Minin119 proved that 2, 8, 9, 12, 18, Sq and 12p (where q 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. Minin120 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 <N and prime to N. M. Lerch121 considered the number x(a, &) and sum X(a, 6) of the divisors ^b of a, proved his100 final formula (17) and c c 2X(ra— an, a) = Sa(x(m— an, ri)—\l/(m — an, a)}, a»l a"! (18) S Mm -an, ~ ) = 2 xfa-an, rn), c^j^1- 1- 0=0 \ ' / a»o L n j If 5 ranges over the divisors of n, "am, n)} =S(«, m; a), n)). m~l He quoted (p. 8) from a letter to him from Chr. Zeller that 2 a\f/(m— a, a) a-l equals the sum of the remainders obtained on dividing m by the integers . M. Lerch122 proved that — crn, n) — S -- p |> \ x» / \ irm"~n rm-f-n~l -p-pn, cr) =Sx(w-p~o-n, n)~" ' 117Casopis, Prag, 21, 1892, 90-95, 185-190 (in Bohemian). Cf. Jahrbuch Fortschritte Math., 24, 1892; 186-7. U8Jour. ffir Math., 112, 1893, 29. "'Math. Soc. Moscow, 17, 1893, 240-253. 120/6td., 17, 1894, 537-544. mPrag Sitzungsberichte (Math.), 1894, No. 11. «2/6td., No. 32.