290 
History of the Theory of Numbers. 
[Chap.X 
where, if n is of the form x{x+l)/2, (t(0) is to be taken to be n/3 [Glai- 
sher^"]. The proof follows from the logarithmic derivative of Jacobi's^" 
expression for s^, as in Euler's^ proof of (2). 
Halphen"*^ formed for an odd function /(z) the sum of s.i 
pc! 
(-1)7 
(t*«)' 
n- 
■ +^n-l — "~o-^n> 
X taking all integral values between the two square roots of a, and y ranging 
over all positive odd divisors of a—x^. This sum is 
if a is a square, zero if a is not a square. Taking /(s) =z, we get a recursion 
formula for the sum of those di\dsors d oi x for which x/d is odd [see the 
topic Sums of Squares in Vol. II of this History]. Taking f{z)=a^ —oT', 
we get a recursion formula for the number of odd di\dsors <a/m of a. 
A generalization of (2) gives a recursion formula for the sima of the divisors 
of the forms 2nk, n{2k-\-\)^m, wdth fixed n, m. 
E. Catalan^'- denoted the square of (1) by l+LiX+ . . . +L„x'*H- . . .. 
Thus 
o-(n) +IiO-(7i- 1) +L20-(n-2) + 
I'n-I>n-l-I>.-2+I^n-54-Z>„_7- . . . = Or (2X + 1)(-1)\ 
according as n is not or is of the form X(X + l)/2. In \dew of the equality 
of (3) and (4) and the fact that l/p=2;/'(n)x", where yp{n) is the number of 
partitions of n into equal or distinct positive integers, he concluded that 
(7(n)=;//(n-l)+2,/'(n-2)-5;/'(n-5)-7i/^(n-7) + 12^(n-12)+. . .". 
J. W. L. Glaisher^^ noted that, if B{n) is the excess of the sum of the odd 
di\'isors of n over the sum of the even dii'isors, 
e{n) +<9(n - 1) +d{n - 3) +d{n - 6) -f . . . = 0, 
where 1, 3, 6, . . . are the triangular numbers, and B{n—n) = —n. 
E. Cesaro^ denoted bj^ s„ the sum of the residues obtained by dividing n 
by each integer <n, and stated that 
s„+(7(l)+(7(2)+...+(7(n)=n2. 
E. Catalan^^ proved the equivalent result that the sum of the divisors of 
1, . . . , n equals the sum of the greatest multiples, not >/?, of these numbers. 
Catalan'*^ stated that, if <^(a, n) is the greatest multiple ^^ of a, 
n 
a{n)= 2 {</)(a, n)—4>{a, n — 1)). 
I 
"Bull. Soc. Math. France, 6, 1877-S, 119-120, 173-188. 
"Assoc, frang. avanc. sc, 6, 1877, 127-8. Cf. Catalan.^^" 
«Messenger Math., 7, 1877-8, 66-7. 
"Nouv. Corresp. Math., 4, 1878, 329; 5, 1879, 22; Nouv. Ann. Math., (3), 2, 1883, 289; 4, 1885, 
473. 
«/Wd., 5, 1879, 296-8; stated, 4, 1879, ex. 447. 
*mid., 6, 1880, 192. 
