Search: id:A103222
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%I A103222
%S A103222 1,1,2,2,2,2,6,4,6,0,10,4,8,6,4,8,12,6,18,0,12,10,22,8,10,4,18,12,22,0,
%T A103222 30,16,20,8,12,12,30,18,16,0,32,12,42,20,12,22,46,16,42,0,24,8,44,18,20,
%U A103222 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 A103222 Real part of the totient function phi(n) for Gaussian integers. See A103223 
               for the imaginary part and A103224 for the norm.
%C A103222 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 A103222 T. D. Noe, <a href="b103222.txt">Table of n, a(n) for n=1..1000</a>
%H A103222 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TotientFunction.html">Totient Function</a>
%F A103222 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 A103222 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 A103222 Sequence in context: A080400 A119462 A096625 this_sequence A061033 A075094 
               A151704
%Y A103222 Adjacent sequences: A103219 A103220 A103221 this_sequence A103223 A103224 
               A103225
%K A103222 nice,nonn
%O A103222 1,3
%A A103222 T. D. Noe (noe(AT)sspectra.com), Jan 26 2005

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