312 History of the Theory of Numbers. [Chap, x 
From the sum of Euler's (f}(d) for the di\'isors d of n, he obtained 
a{n)=iT(jj<t>iJ), nr{n)=i<T(tjct>{j). 
E. Lucas^^- proved the last formulas, the result of Cesaro/* and the 
related one o-(n)+s„ = s„_i+2n — 1. 
A. Berger"^ considered the mean of the number of decompositions of 
1,2,. . ., X into three or more factors, and gave long expressions for \l/{l)-\- 
.. .+^(7J), where i/'(A-) =2d'c?i*', summed for the solutions of ddi = k. He 
gave (pp. 116-125) complicated results on the mean value of (Tk{n). 
D. N. Sokolov and D. T. Egorov^^^" proved, by use of Bougaief's formu- 
las for sums extending over all the divisors of a number, the formulas in 
Liou\'ille's-°"'-^ series of four articles. 
J. W. L. Glaisher^^"* gave Zeller's^^ formula and 
P(n - 1) +2-P(n - 2) - 5-P(n - 5) - 7'P{n - 7) + . . . 
= ^|5(r3(n)-(18n-lMn)l, 
where 1, 2, 5, . . . are pentagonal numbers (3r=i=r)/2 and P(0) = 1. 
Glaisher"^ proved formulae which are greatly shortened by setting 
a./n)=n''(rXn)-3(n-l)V/n-l)+5(n-3)VXn-3)-7(n-6)'(T,(n-6)+.... 
Write Qij for ay(n). Besides the formula [of Halphen^°] aoi = 0, he gave 
40 
ao3-2an = 0, ao5-10ai3+ya2i = 0, 
126 ,756 ^_ . 
007 5~^15H H~^23 — 105031 = 0, 
Oo9 - 50017+720025 - 336O033 +336O041 = 0, 
with the agreement that o-(O) =n/3 and 
— t--\-l f — 1 — ^'*+l t^—1 
^3(0) = -24o-' <^M = -^' ^7(0)— ^^> <^9i0)=-^, 
where t = Sn+l, but did not find the general formula of this type. Next, 
he gave five formulas of another set, the first one being that of his earlier 
paper,^ the second involving the same function of 0-3 with added terms in 
ra{r). Finally, denoting Euler's formula (2) by Ea{n) =0, it is shown that 
5Eas{n)-lSE{n(T{n)\ =0. 
Glaisher^^^ showed that his"^ third formula holds for all odd numbers v 
not expressible as a sum of three squares and hence in particular for the 
"T'htorie des nombres, 1891, 403-6, 374, 388. 
'"Nova Acta Soc. Upsal., (3), 14, 1891 (1886), No. 2, p. 63. 
>"^Math. Soc. Moscow, 16, 1891, 89-112, 236-255. 
"♦Messenger Math., 21, 1891-92, 47-8. 
^''Ibid., 4^-69. 
^*Ibid., 122, 126. The further results are quoted in the chapter on sums of three squares. 
