Chap. X] SuM AND NuMBER OF DiVISORS. 303 
where fx = m or m/2 according as m is odd or even. Cf . Hacks.^^ 
M. A. Stern^^ noted that Zeller's^^ formula follows from B=pA, where 
1 00 n 00 
= A= Si/'(n)a;", -^=S= S o-(n)a:'*-\ p = l+2a:-5x*-7x^+ 
pix) n=o V{x) 
where p(rc) is defined by (1), yp{n) is the number of partitions of n, and 
the second equation follows from the equality of (3) and (4) after remov- 
ing the factor x. Next, if N{n) denotes the number of combinations of 
1, 2, . . ., n without repetitions producing the sum n, 
X N(n)x''= {l+x){l+x^) . . . = 
2, (l-x')il-x') 
rZi ^ ' ' ' '' '^■■- {l-x){l-x')...' 
then by the second equation above, 
B{\-x^-x'^x^''+x^^- . . .)=pSA^(n)x~, 
(r(n)-o-(n-2)-o'(7i-4)+(r(n-10)+(7(n-14)- . . . 
=i\r(n-l)+2i\r(n-2)-5iV(n-5)-7iV(n-7)+..., 
where (T{n — n) =0, N{n—n) = 1. 
S. Roberts^^ noted that Euler's^ formula (2) is identical with Newton's 
relation S^n = S-n+i+S^n+2— ■ ■ ■ for obtaining the sum aS_„ of the (— n)th 
powers of the roots of s = 0, where s and p are defined by (2). In p, the sum 
of the ( — n)th powers of the roots of 1 —a:^ = is A; or according as k is or is 
not a divisor of n. Hence the like sum for p is (r{n). [Cf. Waring^.] The 
process can be applied to products of factors 1 —f(k)x^. His further results 
may be given the following simpler form. Let 0„ be the sum of the even 
divisors of n, and xpn the sum of the odd divisors, and set s„=0„+2i^„ if n 
is even, s„= — 2i^„ if n is odd. By elliptic function expansions, 
S2n + 8{s2n_itAi+3S2H-2'/'2+S2n-3'A3 + 3S2n-4^4+ • • • -hSiXf/ 2n-l] +12ni/^2n = 0, 
S2n+l+8{s2„lAi+3S2„_l^2+ ■ • ■ +3SilA2n) +(4n + 2)l/'2n+l=0» 
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. Bougaief^®" obtained, as special cases of a summation formula, 
^{Sx+5-5i2u-iy}(Ti2x + l-u'' + u) = 0, S{n -3(7(t/)}P{n -o-(w)} = 0, 
where P{n) is the number of solutions u, v of a{u) +(t(v) =n. 
L. Gegenbauer^®'' 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, (nH-3)/6, . . ., 
(2n)/(2n). The number of divisors of the form Bx—'yoil,...,nis ex- 
pressed as a sum of greatest integers; etc. 
J. W. L. Glaisher^^ considered the sum A^n) of the sth powers of the odd 
divisors of n, the Hke sum Dsin) for the even divisors, the sum D',{n) of the 
s^Acta Mathematica, 6, 1885, 327-8. 
8«Quar. Jour. Math., 20, 1885, 370-8. 
8««Comptes Rendus Paris, 100, 1885, 1125, 1160. 
sebDenkschr. Akad. Wiss Wien (Math.), 49, II, 1885, 111. 
8'Messenger Math., 15, 1885-6, 1-20. 
