290 HISTORY OF THE THEORY OF NUMBERS. [CHAP.X where, if n is of the form x(x+i)/2, <r(0) is to be taken to be n/3 [Glai-sher63'67]. The proof follows from the logarithmic derivative of Jacobi's10 expression for s3, as in Euler's5 proof of (2). Halphen41 formed for an odd function /(z) the sum of <-.»(*=*+.). x taking all integral values between the two square roots of a, and y ranging over all positive odd divisors of a— x2. This sum is if a is a square, zero if a is not a square. Taking /(z) =z, we get a recursion formula for the sum of those divisors d of x for which x/d is odd [see the topic Sums of Squares in Vol. II of this History]. Taking f(z) =a* —a""*, we get a recursion formula for the number of odd divisors < a/m of a. A generalization of (2) gives a recursion formula for the sum of the divisors of the forms 2nk, n(2k+l)^=my with fixed n, m. E. Catalan42 denoted the square of (1) by 1+L&+ . . . +Lnxn+. . .. Thus (r(n)+Ll0r(n-"l)+L2cr(n-2)+. . . 5+Ln.7- ... =0 or (2X+1)(-1)X, according as n is not or is of the form X(X+l)/2. In view of the equality of (3) and (4) and the fact that l/p=2ty(n)a;n, where \l/(n) is the number of partitions of n into equal or distinct positive integers, he concluded that J. W. L. Glaisher43 noted that, if 6(n) is the excess of the sum of the odd divisors of n over the sum of the even divisors, 0(w)+0(n-l)+0(n-3)+0(w-6)+...=0, where 1, 3, 6, ... are the triangular numbers, and 0(n—ri) = — n. E. Cesaro44 denoted by sn the sum of the residues obtained by dividing n by each integer <n, and stated that sn+cr(l)+<r(2)+ . . . +CT(TI) =n2. E. Catalan45 proved the equivalent result that the sum of the divisors of 1, . . . , n equals the sum of the greatest multiples, not >n, of these numbers. Catalan46 stated that, if <£(a, n) is the greatest multiple ^n of a, 4lBuU. Soc. Math. France, 6, 1877-8, 119-120, 173-188. 42Assoc. frang. avanc. sc., 6, 1877, 127-8. Cf . Catalan.370 "Messenger Math., 7, 1877-8, 66-7. 44Nouv. Corresp. Math., 4, 1878, 329; 5, 1879, 22; Nouv. Ann. Math., (3), 2, 1883, 289; 4, 1885, 473. «JWd., 5, 1879, 296-8; stated, 4, 1879, ex. 447. "Ibid., 6, 1880, 192.