312 HISTOEY or THE THEORY OF NUMBEBS. [CHAP. X From the sum of Euler's 0(d) for the divisors d of n, he obtained tr(n) =T*O'), nr(n) « E. Lucas112 proved the last formulas, the result of Cesaro,44 and the related one <r(ri) + sn = sn_i + 2n — 1 . A. Berger113 considered the mean of the number of decompositions of 1, 2, . . . , x into three or more factors, and gave long expressions for ^(1) + . ..4-^(n), where \I/(K) =2jdsdil, summed for the solutions of ddi~k. He gave (pp. 116-125) complicated results on the mean value of ^(n). D. N. Sokolov and D. T. Egorov113a proved, by use of Bougaief s formulas for sums extending over all the divisors of a number, the formulas in Liouville's25"29 series of four articles. J. W. L. Glaisher114 gave Zeller's66 formula and P(n-l)+22P(n-2)-52P(tt-5)-72P(n-7) + . . . where 1, 2, 5, ... are pentagonal numbers (3r**=r)/2 and P(0) =1. Glaisher115 proved formulae which are greatly shortened by setting Write dij for a# (n) • Besides the formula [of Halphen40] a0i = 0, he gave 40 a03 ~2an = 0, a05 - 10a13 +y a2i = 0, 126 ,756 aog - 50a17 +720a25 - 3360a33 + 3360a4i = 0, with the agreement that cr(0) =n/3 and where £=8n+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,64 the second involving the same function of o-3 with added terms in r<r(r). Finally, denoting Euler's formula (2) by E<r(ri) =0, it is shown that Glaisher116 showed that his76 third formula holds for all odd numbers v not expressible as a sum of three squares and hence in particular for the 112Th£orie des nombres, 1891, 403-6, 374, 388. 113Nova Acta Soc. UpsaL, (3), 14, 1891 (1886), No. 2, p. 63. 113aMath. Soc. Moscow, 16, 1891, 89-112, 236-255. 1MMessenger Math., 21, 1891-92, 47-8. 11676wf., 49-69. U8/6id.t 122, 126. The further results are quoted in the chapter on sums of three squares.