142 History of the Theory of Numbers. [Chap, v 
G. Oltramare^^ obtained for the sum, sum of squares, sum of cubes, and 
sum of biquadrates, of the integers <7na and relatively prime to a the 
respective values 
^m~a<}>{a), JmV0(a) + (-l)»— a0(a,), ll 
o 
imV</>(«) + (-l)"^a2<A(ai), 
Ttl 111 
6 z-O'O 
where a is the number and Oi the product of the distinct prime factors 
ju, I', . . . of a, while ^(aO = (ju^ — l)(j/^ — 1) . . .. The number of integers <n 
which are prime to a is 4>{a)n/a. 
J. Liou\'ille^^ stated that Gauss' proof of S0(d) =iV may be extended to 
the generalization 
2QWc?) = l*+2*+...+iV*, 
where d ranges over the di\'isors of N. He remarked that Binet's^" results 
are readily proved in various ways. Also, 
e);3w={z>wf. 
N. V. Bougaief^^^ stated that, if ^(n) is the number of distinct prime 
factors of n>\, and ^i(n) is their product, 
also a result quoted below with Gegenbauer's^"° generalization. 
August BUnd^^^ reproduced without reference the formulas and proofs 
by Thacker,^^° and gave 
0,(m)=w'</)o(7n)-^^^w'-Vi(/7O + ('2)^«''-'<A2W- • . . +(-l)'<^.(^0. 
E. Lucas^^^ indicated a proof that 7?<^„_i(x) is given symbolically by 
{x+QY-Q\ where, if n = a°6^ . ., 0, = 5,(l-a'-')(l-5'-0 - • •• Thus, if 
IT is the product of the negatives of the primes a, b, . . . , 
2</)i(x) =x4>{x), 3<t>2{x) =(j>{x) (x~ + hA , 403Ct) =.T(/)(.T)(x2+7r). 
'"Mdmoires de I'lnstitut Nat. Gr^nevois, 4, 1856, 1-10. 
""Comptea Rendus Paris, 44, 1857, 753-4; Jour, de Math., (2), 2, 1857, 393-6. 
i"Nouv. Ann. Math., (2), 13, 1874, 381-3; Bull. Sc. Math. Astr., 10, I, 1876, 18. 
'**Ueber die Potenzsummen der iinter einer Zahl ?« Uegenden und zu ihr relativ primen Zahlen, 
Diss., Bonn, 1876, 37 pp. 
i^^Nouv. Ann. Math., (2), 16, 1877, 159; Throne des nombres, 1891, 394. 
