A227334 -id:A227334 - cp's OEIS frontend

A079458 Number of Gaussian integers in a reduced system modulo n.

1, 2, 8, 8, 16, 16, 48, 32, 72, 32, 120, 64, 144, 96, 128, 128, 256, 144, 360, 128, 384, 240, 528, 256, 400, 288, 648, 384, 784, 256, 960, 512, 960, 512, 768, 576, 1296, 720, 1152, 512, 1600, 768, 1848, 960, 1152, 1056, 2208, 1024, 2352, 800, 2048, 1152, 2704

Offset: 1

Vladeta Jovovic, Jan 14 2003

Number of units in the ring consisting of the Gaussian integers modulo n. - Jason Kimberley, Dec 07 2015

			{1, i, 1+2i, 2+i, 3, 3i, 3+2i, 2+3i} is the set of eight units in the Gaussian integers modulo 4. - _Jason Kimberley_, Dec 07 2015

Jason Kimberley, Table of n, a(n) for n = 1..10000
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Wayback Machine link]
Jonathan M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Catalina Calderón, Jose Maria Grau, A. Oller-Marcén, and László Tóth, Counting invertible sums of squares modulo n and a new generalization of Euler's totient function, Publicationes Mathematicae-Debrecen, Vol. 87 (1-2) (2015), pp. 133-145; arXiv preprint, arXiv:1403.7878 [math.NT], 2014.

Equals four times A218147. - Jason Kimberley, Nov 14 2015
Sequences giving the number of solutions to the equation GCD(x_1^2+...+x_k^2, n) = 1 with 0 < x_i <= n: A000010 (k=1), A079458 (k=2), A053191 (k=3), A227499 (k=4), A238533 (k=5), A238534 (k=6), A239442 (k=7), A239441 (k=8), A239443 (k=9).
Cf. A003557, A007947, A204617, A062570.
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), this sequence ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319445.

A079458 := func)>; // Jason Kimberley, Nov 14 2015

with(GaussInt): seq(GIphi(n), n=1..100);

phi[1]=1;phi[p_, s_] := Which[Mod[p, 4] == 3, p^(2 s - 2) (p^2 - 1), Mod[p, 4] == 1, p^(2 s - 2) ((p - 1))^2, True, 2^(2 s - 1)];phi[n_] := Product[phi[FactorInteger[n][[i, 1]], FactorInteger[n][[i, 2]]], {i, Length[FactorInteger[n]]}];Table[phi[n], {n, 1, 33}] (* José María Grau Ribas, Mar 16 2014 *)
f[p_, e_] := (p - 1)*p^(2*e - 1) * If[p == 2, 1, 1 - (-1)^((p-1)/2)/p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 13 2024 *)

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, r*=2^(2*e-1));
        if(p%4==1, r*=(p-1)^2*p^(2*e-2));
        if(p%4==3, r*=(p^2-1)*p^(2*e-2));
    );
    return(r);
} \\ Jianing Song, Sep 16 2018

Multiplicative with a(2^e) = 2^(2*e-1), a(p^e) = (p^2-1)*p^(2*e-2) if p mod 4=3 and a(p^e) = (p-1)^2*p^(2*e-2) if p mod 4=1.
a(n) = A003557(n)^2 * a(A007947(n)), where a(2)=2, a(p)=(p-1)^2 for prime p=1(mod 4), a(p)=p^2-1 for prime p=3(mod 4), and a(n*m)=a(n)*a(m) for n coprime to m. - Jason Kimberley, Nov 16 2015
From Amiram Eldar, Feb 13 2024: (Start)
Dirichlet g.f.: zeta(s-2) * (1 - 1/2^(s-1)) * Product_{p prime > 2} (1 - 1/p^(s-1) - (-1)^((p-1)/2)*(p-1)/p^s).
Sum_{k=1..n} a(k) = c * n^3 / 3 + O(n^2 * log(n)), where c = (3/4) * Product_{p prime > 2} (1 - 1/p^2 - (-1)^((p-1)/2)*(p-1)/p^3) = (3/4) * A334427 * Product_{p prime == 1 (mod 4)} (1 - 2/p^2 + 1/p^3) = 0.6498027559... (Calderón et al., 2015). (End)
a(n) = A204617(n)*A062570(n). - Ridouane Oudra, Jun 05 2024

A062327 Number of divisors of n over the Gaussian integers.

Original entry on oeis.org

1, 3, 2, 5, 4, 6, 2, 7, 3, 12, 2, 10, 4, 6, 8, 9, 4, 9, 2, 20, 4, 6, 2, 14, 9, 12, 4, 10, 4, 24, 2, 11, 4, 12, 8, 15, 4, 6, 8, 28, 4, 12, 2, 10, 12, 6, 2, 18, 3, 27, 8, 20, 4, 12, 8, 14, 4, 12, 2, 40, 4, 6, 6, 13, 16, 12, 2, 20, 4, 24, 2, 21, 4, 12, 18, 10, 4, 24, 2, 36, 5, 12, 2, 20, 16, 6

Offset: 1

Reiner Martin, Jul 12 2001

Divisors which are associates are identified (two Gaussian integers z1, z2 are associates if z1 = u * z2 where u is a unit, i.e., one of 1, i, -1, -i).
a(A004614(n)) = A000005(n). - Vladeta Jovovic, Jan 23 2003
a(A004613(n)) = A000005(n)^2. - Benedikt Otten, May 22 2013

			For example, 5 has divisors 1, 1+2i, 2+i and 5.

Cf. A027748, A124010.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319442.

a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e                  = 2 * e + 1
   f p e | p `mod` 4 == 1 = (e + 1) ^ 2
         | otherwise      = e + 1
-- Reinhard Zumkeller, Oct 18 2011

a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
                 i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 09 2021

Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* Alonso del Arte, Jan 25 2011 *)
DivisorSigma[0,Range[90],GaussianIntegers->True] (* Harvey P. Dale, Mar 19 2017 *)

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,r*=(2*e+1));
        if(p%4==1,r*=(e+1)^2);
        if(p%4==3,r*=(e+1));
    );
    return(r);
}  \\ Joerg Arndt, Dec 09 2016

Presumably a(n) = 2 iff n is a rational prime == 3 mod 4 (see A045326). - N. J. A. Sloane, Jan 07 2003, Feb 23 2007

Multiplicative with a(2^e) = 2*e+1, a(p^e) = e+1 if p mod 4=3 and a(p^e) = (e+1)^2 if p mod 4=1. - Vladeta Jovovic, Jan 23 2003

A078458 Total number of factors in a factorization of n into Gaussian primes.

Original entry on oeis.org

0, 2, 1, 4, 2, 3, 1, 6, 2, 4, 1, 5, 2, 3, 3, 8, 2, 4, 1, 6, 2, 3, 1, 7, 4, 4, 3, 5, 2, 5, 1, 10, 2, 4, 3, 6, 2, 3, 3, 8, 2, 4, 1, 5, 4, 3, 1, 9, 2, 6, 3, 6, 2, 5, 3, 7, 2, 4, 1, 7, 2, 3, 3, 12, 4, 4, 1, 6, 2, 5, 1, 8, 2, 4, 5, 5, 2, 5, 1, 10, 4, 4, 1, 6, 4, 3, 3, 7, 2, 6, 3, 5, 2, 3, 3, 11, 2, 4, 3, 8, 2, 5, 1, 8

Offset: 1

N. J. A. Sloane, Jan 11 2003

a(n)+1 is also the total number of factors in a factorization of n+n*i into Gaussian primes. - Jason Kimberley, Dec 17 2011

Record high values are at a(2^k) = 2*k for k = 0, 1, 2, ... . - Bill McEachen, Oct 11 2022

			2 = (1+i)*(1-i), so a(2) = 2; 9 = 3*3, so a(9) = 2.
a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 3*a(2)+2*a(3)+4*a(5)+5*a(7)+3*a(11) = 3*2+2*1+4*2+5*1+3*1 = 24. - _Vladeta Jovovic_, Jan 20 2003

T. D. Noe, Table of n, a(n) for n = 1..10000
Michael Somos, PARI program for finding prime decomposition of Gaussian integers.
Eric Weisstein's World of Mathematics, Gaussian Prime.
Index entries for Gaussian integers and primes.

Cf. A078908-A078911, A007814, A065339, A083025.
Cf. A239626, A239627 (including units).
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), A227334 ("psi", A002322), A086275 ("omega", A001221), this sequence ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319444.

Join[{0}, Table[f = FactorInteger[n, GaussianIntegers -> True]; cnt = Total[Transpose[f][[2]]]; If[MemberQ[{-1, I, -I}, f[[1, 1]]], cnt--]; cnt, {n, 2, 100}]] (* T. D. Noe, Mar 31 2014 *)
a[n_]:=PrimeOmega[n, GaussianIntegers -> True]; Array[a,104] (* Stefano Spezia, Sep 29 2024 *)

a(n)=my(f=factor(n)); sum(i=1,#f~,if(f[i,1]%4==3,1,2)*f[i,2]) \\ Charles R Greathouse IV, Mar 31 2014

Fully additive with a(p)=2 if p=2 or p mod 4=1 and a(p)=1 if p mod 4=3. - Vladeta Jovovic, Jan 20 2003

a(n) depends on the number of primes of the forms 4k+1 (A083025) and 4k-1 (A065339) and on the highest power of 2 dividing n (A007814): a(n) = 2*A007814(n) + 2*A083025(n) + A065339(n). - T. D. Noe, Jul 14 2003

More terms from Vladeta Jovovic, Jan 12 2003

A086275 Number of distinct Gaussian primes in the factorization of n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 1, 1, 3, 1, 2, 2, 2, 3, 1, 2, 2, 1, 3, 2, 2, 1, 2, 2, 3, 1, 2, 2, 4, 1, 1, 2, 3, 3, 2, 2, 2, 3, 3, 2, 3, 1, 2, 3, 2, 1, 2, 1, 3, 3, 3, 2, 2, 3, 2, 2, 3, 1, 4, 2, 2, 2, 1, 4, 3, 1, 3, 2, 4, 1, 2, 2, 3, 3, 2, 2, 4, 1, 3, 1, 3, 1, 3, 4, 2, 3, 2, 2, 4, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3

Offset: 1

T. D. Noe, Jul 14 2003

As shown in the formula, a(n) depends on the number of distinct primes of the forms 4*k+1 (A005089) and 4*k-1 (A005091) and whether n is divisible by 2 (A059841).

Note that associated divisors are counted only once. - Jianing Song, Aug 30 2018

			a(1006655265000) = a(2^3*3^2*5^4*7^5*11^3) = 1 + 2*1 + 3 = 6 because n is divisible by 2, has 1 prime factor of the form 4*k+1 and 3 primes of the form 4*k+3. Over the Gaussian integers, 1006655265000 is factored as i*(1 + i)^6*(2 + i)^4*(2 - i)^4*3^2*7^5*11^3, the 6 distinct Gaussian factors are 1 + i, 2 + i, 2 - i, 3, 7 and 11.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Gaussian Prime.

Cf. A005089, A005091, A059841.
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), A227334 ("psi", A002322), this sequence ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319443.

Join[{0}, Table[f=FactorInteger[n, GaussianIntegers->True]; cnt=Length[f]; If[MemberQ[{-1, I, -I}, f[[1, 1]]], cnt-- ]; cnt, {n, 2, 100}]]
a[n_]:=If[n==2,1,PrimeNu[n, GaussianIntegers -> True]]; Array[a,100] (* Stefano Spezia, Sep 29 2024 *)

a(n)=my(f=factor(n)[,1]); sum(i=1,#f,if(f[i]%4==1,2,1)) \\ Charles R Greathouse IV, Sep 14 2015

a(n) = A059841(n) + 2*A005089(n) + A005091(n).

Additive with a(p^e) = 2 if p = 1 (mod 4), 1 otherwise. - Franklin T. Adams-Watters, Oct 18 2006

A317797 Sum of the norm of divisors of n over Gaussian integers, with associated divisors counted only once.

Original entry on oeis.org

1, 7, 10, 31, 36, 70, 50, 127, 91, 252, 122, 310, 196, 350, 360, 511, 324, 637, 362, 1116, 500, 854, 530, 1270, 961, 1372, 820, 1550, 900, 2520, 962, 2047, 1220, 2268, 1800, 2821, 1444, 2534, 1960, 4572, 1764, 3500, 1850, 3782, 3276, 3710, 2210, 5110, 2451, 6727

Offset: 1

Jianing Song, Aug 07 2018

Equivalent of sigma (A000203) in the ring of Gaussian integers. Note that only norms are summed up.

			Let ||d|| denote the norm of d.
a(2) = ||1|| + ||1 + i|| + ||2|| = 1 + 2 + 4 = 7.
a(5) = ||1|| + ||2 + i|| + ||2 - i|| + ||5|| = 1 + 5 + 5 + 25 = 36. Note that 2 - i and 1 + 2i are associated so their norm is only counted once.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Gaussian integer

Cf. A001157.
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), this sequence ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319449.

f[p_, e_] := If[p == 2, 2^(2*e + 1) - 1, Switch[Mod[p, 4], 1, ((p^(e + 1) - 1)/(p - 1))^2, 3, (p^(2 e + 2) - 1)/(p^2 - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 12 2020 *)

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, r*=(2^(2*e+1)-1));
        if(Mod(p,4)==1, r*=((p^(e+1)-1)/(p-1))^2);
        if(Mod(p,4)==3, r*=(p^(2*e+2)-1)/(p^2-1));
    );
    return(r);
}

Multiplicative with a(2^e) = sigma(2^(2e)) = 2^(2e+1) - 1, a(p^e) = sigma(p^e)^2 = ((p^(e+1) - 1)/(p - 1))^2 if p == 1 (mod 4) and sigma_2(p^e) = A001157(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4).

A318608 Moebius function mu(n) defined for the Gaussian integers.

Original entry on oeis.org

1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0

Offset: 1

Jianing Song, Aug 30 2018

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Gaussian prime factor, otherwise (-1)^t if n is a product of a Gaussian unit and t distinct Gaussian prime factors.
a(n) = 0 for even n since 2 = -i*(1 + i)^2 contains a squared factor. For rational primes p == 1 (mod 4), p is always factored as (x + y*i)(x - y*i), x + y*i and x - y*i are not associated so a(p) = (-1)*(-1) = 1.
Interestingly, a(n) and A091069(n) have the same absolute value (= |A087003(n)|), since the discriminants of the quadratic fields Q[i] and Q[sqrt(2)] are -4 and 8 respectively, resulting in Q[i] and Q[sqrt(2)] being two of the three quadratic fields with discriminant a power of 2 or negated (the other one being Q[sqrt(-2)] with discriminant -8).

			a(15) = -1 because 15 is factored as 3*(2 + i)*(2 - i) with three distinct Gaussian prime factors.
a(21) = (-1)*(-1) = 1 because 21 = 3*7 where 3 and 7 are congruent to 3 mod 4 (thus being Gaussian primes).

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Gaussian integer

Absolute values are the same as those of A087003.
First row and column of A103226.
Cf. A101455.
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), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), this sequence ("mu", A008683).
Equivalent in the ring of Eisenstein integers: A319448.
Cf. A091069 (Moebius function over Z[sqrt(2)]).

f[p_, e_] := If[p == 2 || e > 1, 0, Switch[Mod[p, 4], 1, 1, 3, -1]]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 10 2020 *)

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=0);
        if(Mod(p,4)==3&e==1, r*=-1);
    );
    return(r);
}

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - (p mod 4)) where the product is taken over the primes.
Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == 1 (mod 4) and -1 if p == 3 (mod 4).
a(n) = 0 if A078458(n) != A086275(n), otherwise (-1)^A086275(n).
a(n) = A103226(n,0) = A103226(0,n).
For squarefree n, a(n) = Kronecker symbol (-4, n) = A101455(n). Also for these n, a(n) = A091069(n) if n even or n == 1 (mod 8), otherwise -A091069(n).

A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n.

Original entry on oeis.org

1, 3, 6, 6, 24, 6, 6, 12, 6, 24, 120, 6, 12, 6, 24, 24, 288, 6, 18, 24, 6, 120, 528, 12, 120, 12, 18, 6, 840, 24, 30, 48, 120, 288, 24, 6, 36, 18, 12, 24, 1680, 6, 42, 120, 24, 528, 2208, 24, 42, 120, 288, 12, 2808, 18, 120, 12, 18, 840, 3480, 24, 60

Offset: 1

Jianing Song, Sep 19 2018

nonn

Equivalent of psi (A002322) in the ring of Eisenstein integers.
a(n) is the exponent of the multiplicative group of Eisenstein integers modulo n, i.e., (Z[w]/nZ[w])* = {a + b*w : a, b in Z/nZ and gcd(a^2 + a*b + b^2, n) = 1} where w = (1 + sqrt(3)*i)/2. The number of elements in (Z[w]/nZ[w])* is A319445(n).
a(n) is the smallest e such that for any Eisenstein integer z coprime to n we have z^e == 1 (mod n).
By definition, A319445(n)/a(n) is always an integer, and is 1 iff (Z[w]/nZ[w])* is cyclic, that is, rank((Z[w]/nZ[w])*) = A319447(n) = 0 or 1, and n has a primitive root in (Z[w]/nZ[w])*. A319445(n)/a(n) = 1 iff n = 1, 3 or a prime congruent to 2 mod 3.
For n > 2, a(n) is divisible by 6.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Let G = (Z[w]/4Z[w])* = {1, w, 1 + w, w', 1 + w', -1 + 2w, -1, -w, -1 - w, -w', -1 - w', -1 + 2w'}. The possibilities for the exponent of G are 12, 6, 4, 3, 2 and 1. G^6 = {x^6 mod 4 : x belongs to G} = {1} and w^3 !== 1 (mod 4), w^4 !== 1 (mod 4). Therefore, the exponent of G is greater than 4, accordingly the exponent of G is 6 and a(4) = 6.

Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Eisenstein integer
Wikipedia, Torsion group

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): A319442 ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), this sequence ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A227334.

f[p_, e_] := If[p == 3 , If[e == 1, 6, 2*3^(e - 1)], Switch[Mod[p, 3], 1, (p - 1)*p^(e - 1), 2, (p^2 - 1)*p^(e - 1)]]; eisPsi[1] = 1; eisPsi[n_] := LCM @@ f @@@ FactorInteger[n]; Array[eisPsi, 100]  (* Amiram Eldar, Feb 10 2020 *)

a(n)=
{
    my(r=1, f=factor(n));
    for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
        if(p==3, r=lcm(r,2*3^max(e-1,1)));
        if(p%3==1, r=lcm(r,(p-1)*p^(e-1)));
        if(p%3==2, r=lcm(r,(p^2-1)*p^(e-1)));
    );
    return(r);
}

a(3) = 6, a(3^e) = 2*3^(e-1) for e >= 2; a(p^e) = (p - 1)*p^(e-1) if p == 1 (mod 3) and (p^2 - 1)*p^(e-1) if p == 2 (mod 3). If gcd(m, n) = 1 then a(mn) = lcm(a(m), a(n)). [See the group structure of (Z[w]/(pi^e)Z[w])* in A319447, where pi is a prime element in Z[w]. - Jianing Song, Oct 03 2022]

Corrected by Jianing Song, Jan 12 2019

A316506 a(n) is the rank of the multiplicative group of Gaussian integers modulo n.

Original entry on oeis.org

0, 1, 1, 2, 2, 2, 1, 3, 2, 3, 1, 3, 2, 2, 3, 3, 2, 2, 1, 4, 2, 2, 1, 4, 2, 3, 2, 3, 2, 4, 1, 3, 2, 3, 3, 3, 2, 2, 3, 5, 2, 3, 1, 3, 3, 2, 1, 4, 2, 3, 3, 4, 2, 2, 3, 4, 2, 3, 1, 5, 2, 2, 3, 3, 4, 3, 1, 4, 2, 4, 1, 4, 2, 3, 3, 3, 2, 4, 1, 5, 2, 3, 1, 4, 4, 2, 3

Offset: 1

Jianing Song, Jul 05 2018

nonn

The rank of a finitely generated group rank(G) is defined to be the size of the minimal generating sets of G.
Let p be an odd prime and (Z[i]/nZ[i])* be the multiplicative group of Gaussian integers modulo n, then: (Z[i]/p^e*Z[i])* = (C_((p-1)*p^(e-1)))^2 if p == 1 (mod 4); C_(p^(e-1)) X C_(p^(e-1)*(p^2-1)) if p == 3 (mod 4). (Z[i]/2Z[i])* = C_2, (Z[i]/2^e*Z[i])* = C_4 X C_(2^(e-2)) X C_(2^(e-1)) for e >= 2. If n = Product_{i=1..k} (p_i)^(e_i), then (Z[i]/nZ[i])* = (Z[i]/(p_1)^(e_1)*Z[i])* X (Z[i]/(p_2)^(e_2)*Z[i])* X ... X (Z[i]/(p_k)^(e_k)*Z[i])*.
The order of (Z[i]/nZ[i])* is A079458(n) and the exponent of it is A227334(n).
{a(n)} is not additive: (Z[i]/2Z[i])* = C_2, (Z[i]/9Z[i])* = C_3 X C_24, so (Z[i]/18Z[i])* = C_6 X C_24, a(18) < a(2) + a(9). The same problem occurs for a(36), a(54) and a(72) and so on. But note that (Z[i]/63Z[i])* = C_3 X C_24 X C_48 and a(63) = a(7) + a(9).
A079458(n)/A227334(n) is always an integer, and is 1 if and only if (Z[i]/nZ[i])* is cyclic, that is, rank((Z[i]/nZ[i])*) = a(n) = 0 or 1, and n has a primitive root in (Z[i]/nZ[i])*. a(n) = 1 if and only if n = 2 or a prime congruent to 3 mod 4. - Jianing Song, Jan 08 2019
From Jianing Song, Oct 03 2022: (Start)
More generally, let pi be a prime element of Z[i] of norm p or p^2 for prime p, then:
- for p == 1 (mod 4), (Z[i]/(pi^e)Z[i])* = C_((p-1)*p^(e-1));
- for p == 3 (mod 4), (Z[i]/(pi^e)Z[i])* = C_(p^(e-1)) X C_(p^(e-1)*(p^2-1));
- for p = 2, (Z[i]/(pi^e)Z[i])* = C_1 for e = 1, C_2 for e = 2, C_4 X C_(2^floor((e-3)/2)) X C_(2^ceiling((e-3)/2)) for e >= 3.
For a more general result see my link below. (End)

			(Z[i]/1Z[i])* = C_1 (has rank 0);
(Z[i]/2Z[i])* = C_2 (has rank 1);
(Z[i]/3Z[i])* = C_8 (has rank 1);
(Z[i]/4Z[i])* = C_2 X C_4 (has rank 2);
(Z[i]/5Z[i])* = C_4 X C_4 (has rank 2);
(Z[i]/6Z[i])* = C_2 X C_8 (has rank 2);
(Z[i]/7Z[i])* = C_48 (has rank 1);
(Z[i]/8Z[i])* = C_2 X C_4 X C_4 (has rank 3);
(Z[i]/9Z[i])* = C_3 X C_24 (has rank 2);
(Z[i]/10Z[i])* = C_2 X C_4 X C_4 (has rank 3).

Jianing Song, Table of n, a(n) for n = 1..10000
Jianing Song, Structure of (R/P^e)* for R the ring of integers of a quadratic field and P a prime ideal of R

Cf. A046072, A079458, A227334, A302254.

Equivalent in the ring of Eisenstein integers: A319447.

rad(n) = factorback(factorint(n)[, 1]);
grad(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==1, r*=2);
        if(p==2&e==2, r*=4);
        if(p==2&e>=3, r*=8);
        if(p%4==1, r*=(rad(p-1))^2);
        if(p%4==3&e==1, r*=rad(p^2-1));
        if(p%4==3&e>=2, r*=p^2*rad(p^2-1));
    );
    return(r);
}
a(n)=if(n>1, vecmax(factor(grad(n))[, 2]), 0);

Let p be an odd prime, then: a(p^e) = 2 if p == 1 (mod 4) or p == 3 (mod 4), e >= 2; a(p) = 1 if p == 3 (mod 4). a(2) = 1, a(4) = 2, a(2^e) = 3 for e >= 3.

A302254 Exponent of the group of the Gaussian integers in a reduced system modulo (1+i)^n.

Original entry on oeis.org

1, 1, 2, 4, 4, 4, 4, 4, 8, 8, 16, 16, 32, 32, 64, 64, 128, 128, 256, 256, 512, 512, 1024, 1024, 2048, 2048, 4096, 4096, 8192, 8192, 16384, 16384, 32768, 32768, 65536, 65536, 131072, 131072, 262144, 262144, 524288, 524288, 1048576, 1048576, 2097152, 2097152, 4194304, 4194304, 8388608, 8388608

Offset: 0

Jianing Song, Apr 04 2018

For n > 0, the number of elements in the group of the Gaussian integers in a reduced system modulo (1+i)^n is 2^(n-1).

			For Gaussian integer x such that (x, 1+i) = 1, x^4 - 1 = (x + 1)(x - 1)(x + i)(x - i) provides at least 7 factors of 1+i in total (and exactly 7 when x = 2+i), so a(7) = 4.

Cf. A016116, A079458, A227334.

[1,1,2,4,4,4] cat [2^(Floor(n div 2)-1): n in [6..50]]; // Vincenzo Librandi, Apr 04 2018

Join[{1, 1, 2, 4, 4, 4}, Table[2^(Floor[n/2] - 1), {n, 6, 50}]] (* Vincenzo Librandi, Apr 04 2018 *)

For n > 5, a(n) = 2^(floor(n/2) - 1).

For even n, a(n) = A227334(2^(n/2)).

cp's OEIS Frontend

A079458 Number of Gaussian integers in a reduced system modulo n.

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A062327 Number of divisors of n over the Gaussian integers.

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A078458 Total number of factors in a factorization of n into Gaussian primes.

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A086275 Number of distinct Gaussian primes in the factorization of n.

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A317797 Sum of the norm of divisors of n over Gaussian integers, with associated divisors counted only once.

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A318608 Moebius function mu(n) defined for the Gaussian integers.

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A319446 Exponent of the group of the Eisenstein integers in a reduced system modulo n.

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