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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,