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 with different terms from 2, 4, 6, . . . , first taking an even number of the latter and second an odd number, the difference of the counts is 1 or 0 according as n is a triangular number or not. It is proved that (10) The fact that the second member must be an integer is generalized as follows: for n odd, a(n) is even or odd according as n is not or is a square; for n even, <r(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. Bouniakowsky20 proved that <r(AT)s2 (mod 4) only when N=kc2 or 2fcc2, where k is a prime 4Z+1 [corrected by Liouville30]. V. A. Lebesgue21 denoted by 1+A&+ 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 This recursion formula gives — 3) A — w(m— l)(m— 8) -t - 1 The expression for Ak was not found. E. Meissel22 proved that (cf. Dirichlet17) (11) T(n) = / J. Liouville23 noted that by taking the derivative of the logarithm of each member of (6) we get the formula, equivalent to (8) : summed for m — 0, 1,..., the argument of <r remaining ^ 0. J. Liouville24 stated that it is easily shown that *>M6m. Ac. Sc. St. Pftersbcmrg, (6), 5, 1853, 303-322. 21Nouv. Ann. Math., 12,1853, 232-4. ^Jour. fiir Math., 48, 1854, 306. MJour. de Math., (2), 1, 1856, 349-350 (2, 1857, 412). **/6wl, (2), 2, 1857, 56; Nouv. Ann. Math., 16,1857,181; proof by J. J. Hemming, ibid, (2), 4 1865, 547.