CHAP. X] SUM AND NUMBER OF DlVISORS. 309 In particular, if the Pi include all the primes in order, we may replace N(x) by [x]} the greatest integer fSz. Since there are as many divisors >a of n as there are divisors <n/a, where e = 1 or 0 according as n is or is not a divisor of p. These two formulas serve as recursion formulas for the computation of N(ri). For the case of two primes Pi = 2, p2 = 3, The functions L satisfy similar formulas and are computed similarly. J. W. L. Glaisher108 stated a theorem, which reduces for m = l to Halphen's,40 . • .+0m+1), provided m is odd, where k ranges over the even numbers 2, 4, . . ., w-1, while 5 = 0 or 5 = 1 according as n is not or is of the form g(g+l)/2. As in Glaisher67 for ra = l, the series are stopped before any term cr^n—n) is reached; but, if we retain such terms, we must set 5 = 0 for every n and define <rt-(0) by where BI, B2,. . . are the Bernoullian numbers. Glaisher109 stated the simpler generalization of Halphen40: CO where the summation index /c ranges over the even numbers 2, 4, . . . , m — 1, and m is odd. If we include the terms <r2r_i(0) = ( — I)rj3r/(4r) in the left member, the right member is to be replaced by . 2"»+2(m+2) '"Messenger Math., 20, 1890-1, 129-135. »»;&«., 177-181.