284 History of the Theory of Numbers. [Chap, x 
where the exponents in the series are triangular numbers. Hence if we 
count the number of ways in which n can be formed as a sum of different 
terms from 1, 2, 3, . . . together w^ith different terms from 2, 4, 6, . . ., first 
taking an even number of the latter and second an odd number, the differ- 
ence of the counts is 1 or according as n is a triangular number or not. 
It is proved that 
(10) <r(n) + {(T(2)-4o-(l))(r(n-2)+(r(3)(T(n-4) + {(r(4)-4(r(2))(r(n-6) 
+(r(5)(7(n-8) + {(r(6)-4(r(3))tr(n-10)+. . . =^(7(n+2). 
The fact that the second member must be an integer is generaUzed as 
follows: for n odd, (T(n) is even or odd according as n is not or is a square; 
for n even, (T{n) is even if n is not a square or the double of a square, odd in the 
contrary case. Hence squares and their doubles are the only integers whose 
sums of divisors are odd. 
V. Bouniakowsky-'^ proved that (r(A^) = 2 (mod 4) only when N = kc^ or 
2kc^, where A: is a prime 4Z+1 [corrected by Liouville^°]. 
V. A. Lebesgue-^ denoted by l-{-AiX+A2X^-\- . . . the expansion of the 
mth power of p{x), given by (1), and proved, by the method used by Euler 
for the case m = 1, that 
a{n)+A,a{n-l)-\-A2<T{n-2)+ . . .+Ar,_,a{l)-\-nAjm = 0. 
This recursion formula gives 
. m(m— 3) . —m(m — l)(m — S) 
A,= -m, A, = — ^^2— ' ^^ = 1:2:3 •••• 
The expression for Aj, was not found. 
E. MeisseP2 proved that (c/. Dirichlet^^) 
(11) T{n) = i^[jj =^i:[j] -'' (^ = [V^])- 
J. Liouville^^ noted that by taking the derivative of the logarithm of 
each member of (6) we get the formula, equivalent to (8) : 
J 5m(m+l) 1 /o , 1 2 N n 
S^n ^ Ya{2n+l—m—m)=0, 
summed for m = 0, 1, . . ., the argument of a remaining ^0. 
J. Liouville^^ stated that it is easily shown that 
Sd<T(d)=s(|y(r(d), 
20M6m. Ac. Sc. St. P^tersbourg, (6), 5, 1853, 303-322. 
"Nouv. Ann. Math., 12, 1853, 232-4. 
"Jour, fur Math., 48, 1854, 306. 
"Jour, de Math., (2), 1, 1856, 349-350 (2, 1857, 412). 
^Ibid., (2), 2, 1857, 56; Nouv. Ann. Math., 16, 1857, 181; proof by J. J. Hemming, ibid., (2), 4, 
1865, 547. 
