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146                     HISTOKY or THE THEOBY OF NUMBEHS.               [CHAP, v
Hence if a ranges over the integers ^n and prime to n, S(a nJS)p=0 or a multiple of n\l/p according as p is odd or even.   By this recursion formula,
L. Gegenbauer173 gave a formula including those of Nazimov167 and Zsigmondy.77   For any functions x(d), XiW), /(^i,    > st),
where d ranges over all divisors of n which have some definite property P, while 5 ranges over those common divisors of n, xlt . . . , xs which have property P. Various special choices are made for x> Xi> / and P. For instance, property P may be that d is an exact pth power, whence, if p = 1, d is any divisor of n. The special results obtained relate mainly to new number-theoretic functions without great interest and suggested apparently by the topic in hand.
T. del Beccaro174 noted that $k(ri) is divisible by n if k is odd [Binet161]. When n is a power of 2,
l*+2*+. . .+(w-l)fcssO or<(n) (mod n),
according as k is odd or even.   His proof of (1) is due to Euler.
J. W. L. Glaisher175 proved that, if a, 6, ... are any divisors of x such that their product is also a divisor, the sum of the nth powers of the integers < x and not divisible by a or b, . . . , is
where s is the number of the divisors a, b, . . . , and
If a, 6, ... are all the prime factors of x, this result becomes Thacker's.150 N. Nielsen176 proved by induction on 7 that the sum of the nth powers of the positive integers <mM and prime to M  p{1 . . .p^y is
The case m= 1 gives Thacker's150 result. That result shows (ibid., p. 179) that <j>2n(m) and <2+i(X) are divisible by m and m2 respectively, for l^n 2g (pi  3)/2, where pi is the least prime factor of m, and also gives the residues of the quotients modulo m. Corresponding theorems therefore hold for the sum of the products of the integers <ra and prime to m, taken t at a time.
173Sitzungsberichte Ak. Wiss. Wien (Math.), 102, 1893, Ha, 1265-94. 174Atti R. Accad. Lincei, Mem. Cl. Fis. Mat., 1, 1894, 344-371. 176Messenger Math., 28, 1898-9, 39-41. 1760versigt Daaske Vidensk. Selsk. Forhandlinger, 1915, 509-12; cf. 178-9.