A391111 - OEIS

A391111

Carmichael's lambda function on Gaussian integers: least universal exponent for a reduced system of residues modulo a + b*sqrt(-1).

0, 1, 1, 2, 4, 4, 8, 4, 12, 4, 16, 8, 4, 4, 12, 28, 4, 16, 8, 36, 4, 40, 8, 48, 20, 12, 52, 28, 60, 4, 12, 16, 8, 72, 36, 4, 24, 40, 16, 88, 8, 96, 48, 4, 100, 12, 52, 108, 112, 28, 24, 120, 60, 100, 4, 12, 16, 136, 8, 28, 72, 36, 148, 16, 156, 4, 24, 40, 12, 16, 172

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OFFSET

1,4

COMMENTS

In what follows, i = sqrt(-1). This sequence is the canonical generalization of the reduced totient function to imaginary quadratic field Q(i). For Gaussian integers alpha = a + b*i = A229140(n) + A385236(n)*i the exponent lambda(alpha),- the least integer k such that beta^k == 1 (mod alpha) for all beta coprime with alpha,- is computed by factoring norm N_alpha = a^2 + b^2 = A001481(n) into rational primes and intersecting

(1) for ramified prime 2: exponent lambda(2^k) = 2^(k-1) if k is 3, 2^[k/2] if k is 1,2,4 or 5, and 2^([k/2]-1) for k >= 6,

(2) inert primes p = 4n + 3: lambda(p^k) = p^(k+1) - p^(k-1),

(3) split primes q = 4n + 1: lambda(q^k) = q^k - q^(k-1),

with k the multiplicity of each prime element in alpha. That is, apply the cyclic subgroup structure for irreducible elements in quadratic ring Z[i] as summarized in Lassak and Porubsky, p. 277-279. The rational and quadratic exponents then differ for 25 of the 87 Gaussian integers A229140(n) + A385236(n)*i.

LINKS

Table of n, a(n) for n=1..71.

Miroslav Lassak and Stefan Porubsky, Fermat-Euler theorem in algebraic number fields, Journal of Number Theory, Vol. 60, Issue 2 (1996), pp. 254-290.

FORMULA

a(n) = lambda(A229140(n) + A385236(n)*i).

EXAMPLE

a(64) = 16, since A229140(64) + A385236(64)*i = 3 + 12*i with norm 153, which in Z[i] factors as 3*(1 + 4*i), giving exponent lcm(8, 16).

PROG

(PARI) a(n) = {my(Qi = bnfinit(t^2+1)); print1("0, ");

for(n=1, 173, my(f = bnfisintnorm(Qi, n));

if(f, my(v, a=-1, b); until(issquare(n-a*a, &b), a++);

v = idealstar(Qi, [a, b]~, 0)[2];