

A227334


Exponent of the group of the Gaussian integers in a reduced system modulo n.


6



1, 2, 8, 4, 4, 8, 48, 4, 24, 4, 120, 8, 12, 48, 8, 8, 16, 24, 360, 4, 48, 120, 528, 8, 20, 12, 72, 48, 28, 8, 960, 16, 120, 16, 48, 24, 36, 360, 24, 4, 40, 48, 1848, 120, 24, 528, 2208, 8, 336, 20, 16, 12, 52, 72, 120, 48, 360, 28, 3480, 8, 60, 960, 48, 32
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OFFSET

1,2


COMMENTS

a(n) is the exponent of the multiplicative group of Gaussian integers modulo n, i.e., (Z[i]/nZ[i])* = {a + b*i: a, b in Z/nZ and gcd(a^2 + b^2, n) = 1}. The number of elements in (Z[i]/nZ[i])* is A079458(n).
For n > 2, a(n) is divisible by 4.  Jianing Song, Aug 29 2018
Equivalent of psi (A002322) in the ring of Gaussian integers.
a(n) is the smallest positive e such that for any Gaussian integer z coprime to n we have z^e == 1 (mod n).
By definition, A079458(n)/a(n) is always an integer, and is 1 iff (Z[i]/nZ[i])* is cyclic, that is, rank((Z[i]/nZ[i])*) = A316506(n) = 0 or 1, and n has a primitive root in (Z[i]/nZ[i])*. A079458(n)/a(n) = 1 iff n = 1, 2 or a prime congruent to 3 modulo 4. (End)


LINKS



FORMULA

a(2^e) = 2^e if e <= 2 and 2^(e1) if e >= 3, a(p^e) = (p  1)*p^(e1) if p == 1 (mod 4) and (p^2  1)*p^(e1) if p == 3 (mod 4). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)).  Jianing Song, Aug 29 2018 [See the group structure of (Z[i]/(pi^e)Z[i])* in A316506, where pi is a prime element in Z[i].  Jianing Song, Oct 03 2022]


EXAMPLE

Let G = (Z[i]/4Z[i])* = {i, 3i, 1, 1 + 2i, 2 + i, 2 + 3i, 3, 3 + 2i}. The possibilities for the exponent of G are 8, 4, 2 and 1. G^4 = {x^4 mod 4 : x belongs to G} = {1} and i^2 !== 1 (mod 4). Therefore, the exponent of G is greater than 2, accordingly the exponent of G is 4 and a(4) = 4.


MATHEMATICA

fa = FactorInteger; lamas[1] = 1; lamas[p_, s_]:= Which[Mod[p, 4]==3, p^(s1)(p^2  1), Mod[p, 4] == 1, p^(s  1)(p  1), s ≥ 4, 2^(s  1), s > 1, 4, s == 1, 2]; lamas[n_] := {aux = 1; Do[aux = LCM[aux, lamas[fa[n][[i, 1]], fa[n][[i, 2]]]], {i, 1, Length@fa[n]}]; aux}[[1]]; Table[lamas[n], {n, 100}]


PROG

(PARI) a(n)=
{
my(r=1, f=factor(n));
for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
if(p==2&&e<=2, r=lcm(r, 2^e));
if(p==2&&e>=3, r=lcm(r, 2^(e1)));
if(p%4==1, r=lcm(r, (p1)*p^(e1)));
if(p%4==3, r=lcm(r, (p^21)*p^(e1)));
);
return(r);


CROSSREFS

Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), this sequence ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).


KEYWORD

nonn


AUTHOR



STATUS

approved



