Chap. XIX] 
Inversion; Function fx{n). 
445 
P. Bachmann^° proved that f{n) ='Zlz^F{kn) implies that 
F(l) 
S ii{n)f{n). 
n=l 
Write X = [x/n\. Taking F{n) =X, nX, <J>(Z), whence /(n) = T{X), no-(X), 
D{X), respectively, we obtain Lipschitz's^° (Ch. X) formulas: 
ti 
[x]= 2 iJi{n)T\-\=2fji{n)na 
n 
*[a:]=2/x(??)i) 
[Q 
Let F{n) be zero if n is not a divisor of P and write x{/{P/n) for /^(n). 
Hence if d divides P, f{d)=7:4^{P/kd) implies iA(P) =Sm(^)/(^), where 
A; ranges over the divisors of P/d, and d over those of P. 
D. von Sterneck^^ considered a function f(n) with the properties: 
(i) /(I) = 1 ; (ii) the g. c. d. of /(m) and/(n) isf{d) if rf is the g. c. d. of m and n; 
(iii) for primes p, other than specified ones, one of the numbers /(p±l) is 
divisible by p; (iv) the g. c. d. oi f{pn)/f{n) and /(n) divides p. Then if 
L{n) is the 1. c. m. of the values of / for all the divisors <n of n, F{n) =f{n) 
■i-L{n) is an integer which can be given the form 
Fin) = 
^\PrpJ ^\p,PiPkVl)' 
W \pmpk}'" 
n=Jlpi 
The four properties hold for the function defined by the recursion 
formula /(n) =a/(n — l)+i3/(n — 2), where a and jS are relatively prime, 
with the initial conditions /(I) = 1 , /(2) = a. For a = 2x,p = b — x^, we have^^ 
(x+Vbr-ix-VbT 
m= 
2\/b 
The case a = j3 = l was discussed by Lucas-^ of Ch. XVII, and his test for 
primality holds for the present generalization. 
The four properties hold also for 
Kn) = 
aJ'-lf 
if a, h are relatively prime ;^^ then fip — l) is divisible by p if p is a prime 
not dividing a, b or a — b. 
K. Zsigmondy-^ gave a generalized inversion formula. Let A^ be any 
multiple of the relatively prime integers ni, . . . , n^. Set 
where d ranges over those divisors ^m oi N which are products of powers 
2"Die Analytische Zahlentheorie, 1894, 310. 
"Monatshefte Math. Phys., 7, 1896, 37, 342. 
22DirichIet, Werke, 1, 47-62. See Dirichlet,^ Ch. XVII. 
wZsigmondy, Monatshefte Math. Phys., 3, 1892, 265. 
^Ibid., 7, 1896, 190-3. 
