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A103222 Real part of the totient function phi(n) for Gaussian integers. See A103223 for the imaginary part and A103224 for the norm. 6


%S 1,1,2,2,2,2,6,4,6,0,10,4,8,6,4,8,12,6,18,0,12,10,22,8,10,4,18,12,22,

%T 0,30,16,20,8,12,12,30,18,16,0,32,12,42,20,12,22,46,16,42,0,24,8,44,

%U 18,20,24,36,16,58,0,50,30,36,32,8,20,66,16,44,0,70,24,62,24,20,36,60,8,78,0

%N Real part of the totient function phi(n) for Gaussian integers. See A103223 for the imaginary part and A103224 for the norm.

%C This definition of the totient function for Gaussian integers preserves many of the properties of the usual totient function: (1) it is multiplicative: if gcd(z1,z2)=1, then phi(z1*z2)=phi(z1)*phi(z2), (2) phi(z^2)=z*phi(z), (3) z=Sum_{d|z} phi(d) for properly selected divisors d and (4) the congruence z=1 (mod phi(z)) appears to be true only for Gaussian primes. The first negative term occurs for n=130=2*5*13, the product of the first three primes which are not Gaussian primes.

%H Amiram Eldar, <a href="/A103222/b103222.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>

%F Let a nonzero Gaussian integer z have the factorization u p1^e1...pn^en, where u is a unit (1, i, -1, -i), the pk are Gaussian primes in the first quadrant and the ek positive integers. Then we define phi(z) = u*product_{k=1..n} (pk-1) pk^(ek-1).

%t phi[z_] := Module[{f, k, prod}, If[Abs[z]==1, z, f=FactorInteger[z, GaussianIntegers->True]; If[Abs[f[[1, 1]]]==1, k=2; prod=f[[1, 1]], k=1; prod=1]; Do[prod=prod*(f[[i, 1]]-1)f[[i, 1]]^(f[[i, 2]]-1), {i, k, Length[f]}]; prod]]; Re[Table[phi[n], {n, 100}]]

%Y Cf. A103223, A103224.

%K nice,sign

%O 1,3

%A _T. D. Noe_, Jan 26 2005

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