A394288 Values taken by the analog of Euler's totient function for Gaussian integers.

1, 2, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 48, 52, 56, 60, 64, 72, 80, 88, 96, 100, 104, 108, 112, 120, 128, 136, 144, 148, 156, 160, 172, 176, 180, 192, 196, 200, 208, 216, 224, 228, 232, 240, 256, 268, 272, 276, 280, 288, 292, 296, 312, 316, 320, 336, 344, 348, 352, 360, 372, 384

Offset

1,2

Comments

Equivalently, numbers k such that there exists a unit group (Z[i]/(x + yi))* of order k.

For any complex number z, the analog of the Euler product formula for Gaussian integers is N(z)*Product_{p|z} 1 - 1/N(p), where p is a Gaussian prime and N(z) is the norm of z.

Under a multiplication which is not completely multiplicative, the values of Euler's totient function for rationals can be obtained by taking the products of the set {(p-1)*p^(e-1)} over all primes p. The present sequence is generated from the set {(p-1)*p^(r-1), (q^2-1)*(q^2)^(s-1)} = {phi(p^r), J_2(q^s)}, where J_2 is Jordan's totient function, for primes p and q such that p = 2, p == 1 mod 4, and q == 3 mod 4.

The above rules can be generalized to quadratic integer rings which are UFDs: in this case the generating set {(p-1)*p^(r-1), (q^2-1)*(q^2)^(s-1)} is taken over primes p and q where p is either a ramified or split prime, and q is inert. See the Englezou link for a PARI program. Note that the program lst_d(lim, d) generates a subset of the set of totient values of Z[sqrt(d)] for all nonzero integers d; it is only for those d such that Z[sqrt(d)] is a UFD that this subset is equal to the full set.

Links

Miles Englezou, Table of n, a(n) for n = 1..10000

Miles Englezou, PARI program for general quadratic integer rings

Example

2 is a term since for z = 2 = -i(1 + i)^2, N(-i(1 + i)^2)(1 - 1/N(1 + i)) = 4 * 1/2 = 2.
4 is a term since for z = 2 + i, N(2 + i)(1 - 1/N(2 + i)) = 5 * 4/5 = 4.
8 is a term since for z = 4 - 2i = (2 + i)(1 + i)^2, N(4 + 2i)(1 - 1/N(2 + i))(1 - 1/N(1 + i)) = 20 * 4/5 * 1/2 = 8.

Prog

(PARI) lst(lim) = my(S = Set([1])); forprime(p=2, lim, my(L = List()); if(p==2, for(e=1, oo, my(v = 2^(e-1)); if(v > lim, break); listput(L, v)), if(p%4==1, for(e=1, oo, my(v = p^(e-1)*(p-1)); if(v > lim, break); listput(L, v)), for(e=1, oo, my(v = (p^2)^(e-1)*(p^2-1)); if(v > lim, break); listput(L, v)))); my(T = Set(S)); for(i=1, #S, for(j=1, #L, my(v = S[i]*L[j]); if(v <= lim, T = setunion(T, Set([v]))))); S = T); S

Crossrefs

Cf. A002202, A007434, A079458, A391110.

Sequence in context: A019442 A048166 A387056 * A360013 A335738 A364056

Adjacent sequences: A394285 A394286 A394287 * A394289 A394290 A394291

Keyword

nonn

Author

Miles Englezou, Apr 13 2026

Status

approved