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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,
where 1, 3, 6, ... are the triangular numbers, and 0(nri) =  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,
JWd., 5, 1879, 296-8; stated, 4, 1879, ex. 447. "Ibid., 6, 1880, 192.