CHAP. X] SUM AND NUMBER OF DlVISORS. 303
where v = m or m/2 according as m is odd or even. Cf. Hacks.96
M. A. Stern85 noted that Zeller's66 formula follows from B=pA, where
r
P\X)
where p(o;) is defined by (1), \l/(n) is the number of partitions of n, and the second equation follows from the equality of (3) and (4) after removing the factor x. Next, if N(ri) denotes the number of combinations of 1, 2, . . ., n without repetitions producing the sum n,
then by the second equation above,
B(l-x2-x*+x10+xu- . . .)=
where <r(n— n) =0, N(n—ri) = 1.
S. Roberts86 noted that Euler's* formula (2) is identical with Newton's relation >S_n = ^_n+i+^«n+2-" • - - for obtaining the sum £_n of the (— n)th powers of the roots of 5 = 0, where s and p are defined by (2) . In p, the sum of the (— n)th powers of the roots of 1 —xk = 0 is k or 0 according as k is or is not a divisor of n. Hence the like sum for p is cr(ri). [Cf. Waring9.] The process can be applied to products of factors 1 —f(k)xk. His further results may be given the following simpler form. Let 0n be the sum of the even divisors of n, and ^n the sum of the odd divisors, and set sn=<£n+2^n if n is even, sn= — 2i^n if n is odd. By elliptic function expansions,
the coefficients being 1 and 3 alternately. He indicated a process for finding a recursion formula involving the sums of the cubes of the even divisors and the sums of the cubes of the odd divisors, but did not give the formula. N. V. Bougaief86a obtained, as special cases of a summation formula, 2{Sx-\-5-5(2u-iy}a(2x+l-u2 + u) = 0, S{n - 3<r(u)}P{n -er(w)} =0,
where P(ri) is the number of solutions u, v of cr(u) -\-a(v) — n.
L. Gegenbauer866 proved that the number of odd divisors of 1, 2, . . ., n equals the sum of the greatest integers in (n-\-l)/2, (n+2)/4, (n-f3)/6, . . ., (2n)/(2n). The number of divisors of the form Bx—y of 1,. . ., n is expressed as a sum of greatest integers; etc.
J. W. L. Glaisher87 considered the sum Aa(n) of the sth powers of the odd divisors of n, the like sum Da(n) for the even divisors, the sum D'a(ri) of the
^Acta Mathematica, 6, 1885, 327-8. 8fiQuar. Jour. Math., 20, 1885, 370-8. 8fi«Comptcs Rendus- Paris, 100, 1885, 1125, 1160. ^Denkschr. Akad. Wiss Wien (Math.), 49, II, 1885, 111. "Messenger Math., 15, 1885-6, 1-20.