142 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v
G. Oltramare153 obtained for the sum, sum of squares, sum of cubes, and sum of biquadrates, of the integers <ma and relatively prime to a the respective values
where a is the number and ax the product of the distinct prime factors ju, v, ... of a, while J~(ai) — (ju3 — 1) (v3 — 1) . . . . The number of integers <n which are prime to a is <j>(a)n/a.
J. Liouville164 stated that Gauss' proof of 2J0(d) =N may be extended to the generalization
where d ranges over the divisors of N. He remarked that Binet's151 results are readily proved in various ways. Also,
N. V. Bougaief165 stated that, if £(n) is the number of distinct prime factors of n>l, and ^(ri) is their product,
also a result quoted below with Gegenbauer's170 generalization.
August Blind156 reproduced without reference the formulas and proofs by Thacker,150 and gave
E. Lucas157 indicated a proof that n<£n_i(aO is given symbolically by (x+Q)n-Qn, where, if n = aatf. .., Qfc = Bk(l -ak-l)(l-bh~l) . . .. Thus, if TT is the product of the negatives of the primes a, 6, . . . ,
1B3M6moires de 1'Institut Nat. Gdnevois, 4, 1856, 1-10.
1S4Comptes Rendus Paris, 44, 1857, 753-4; Jour, de Math., (2), 2, 1857, 393-6.
15SNouv. Ann. Math., (2), 13, 1874, 381-3; Bull. Sc. Math. Astr., 10, I, 1876, 18.
166Ueber die Potenzsummen der unter einer Zahl m liegendcn und zu ihr relativ primen Zahlcn,