A318608 -id:A318608 - 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

A227334 Exponent of the group of the Gaussian integers in a reduced system modulo n.

Original entry on oeis.org

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

Offset: 1

José María Grau Ribas, Jul 07 2013

nonn

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
From Jianing Song, Sep 23 2018: (Start)
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)

			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.

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

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).

Equivalent in the ring of Eisenstein integers: A319446.

fa = FactorInteger;lamas[1] = 1;lamas[p_, s_]:= Which[Mod[p, 4]==3,p^(s-1)(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}]

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^(e-1)));
        if(p%4==1, r=lcm(r,(p-1)*p^(e-1)));
        if(p%4==3, r=lcm(r,(p^2-1)*p^(e-1)));
    );
    return(r);
} \\ Jianing Song, Aug 29 2018

a(2^e) = 2^e if e <= 2 and 2^(e-1) if e >= 3, a(p^e) = (p - 1)*p^(e-1) if p == 1 (mod 4) and (p^2 - 1)*p^(e-1) 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]

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).

A319448 Moebius function mu(n) defined for the Eisenstein integers.

Original entry on oeis.org

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

Offset: 1

Jianing Song, Sep 19 2018

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Eisenstein prime factor, otherwise (-1)^t if n is a product of an Eisenstein unit and t distinct Eisenstein prime factors.

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2. a(n) = 0 for n divisible by 3 since 3 = w'*(1 + w)^2 contains a squared factor. For rational primes p == 1 (mod 3), p is always factored as (x + y*w)(x + y*w'), x + y*w and x + y*w' are not associated so a(p) = (-1)*(-1) = 1.

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
a(14) = -1 because 14 is factored as 2*(2 + w)*(2 + w') with three distinct Eisenstein prime factors.
a(55) = (-1)*(-1) = 1 because 55 = 5*11 where 5 and 11 are congruent to 2 mod 3 (thus being Eisenstein primes).

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

Cf. A102283.
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), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), this sequence ("mu", A008683).
Equivalent in the ring of Gaussian integers: A318608.

f[p_, e_] := If[p == 3 || e > 1, 0, Switch[Mod[p, 3], 1, 1, 2, -1]]; eisMu[1] = 1; eisMu[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisMu, 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||e>=2, r=0);
        if(Mod(p, 3)==2&e==1, r*=-1);
    );
    return(r);
}

a(n) = 0 if n is divisible by 3 or has a square prime factor, otherwise Product_{p divides n} (3 - 2*(p mod 3)) where the product is taken over the primes.
Multiplicative with a(p^e) = 0 if p = 3 or e > 1, a(p) = 1 if p == 1 (mod 3) and -1 if p == 2 (mod 3).
For squarefree n, a(n) = Legendre symbol (n, 3) = Kronecker symbol (-3, n) = A102283(n).

A332472 The real part of the sum of unitary divisors function (usigma) generalized for Gaussian integers.

Original entry on oeis.org

1, 1, 4, -3, 4, 4, 8, 1, 10, -12, 12, -12, 6, 8, 16, 17, 6, 10, 20, -12, 32, 12, 24, 4, -24, -30, 28, -24, 8, -48, 32, 1, 48, -38, 32, -30, 8, 20, 24, 68, 10, 32, 44, -36, 40, 24, 48, 68, 50, -40, 24, -18, 10, 28, 48, 8, 80, -64, 60, -48, 12, 32, 80, -63, -120

Offset: 1

Amiram Eldar, Feb 13 2020

sign

If n = u * Product_{i} p_i^e_i, where u is a unit (1, i, -1 or -i), and p_i is a Gaussian prime with Re(p_i) > 0, then usigma(n) = Product_{i} (p_i^e_i + 1).

a(n) = A103228(n) for odd squarefree numbers (A056911), i.e., numbers n such that A318608(n) != 0.

			a(4) = -3 since 4 = -(1 + i)^4 in Gaussian integers (i is the imaginary unit), so usigma(4) = (1 + i)^4 + 1 = -3, and a(4) = Re(-3) = -3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A034448, A103228, A332473 (the imaginary part), A332474 (the norm).

f[p_, e_] := If[Abs[p] == 1, 1, (p^e + 1)]; usigma[n_] := Times @@ f @@@ FactorInteger[n, GaussianIntegers -> True]; a[n_] := Re[usigma[n]]; Array[a, 100]

A091069 Moebius mu sequence for real quadratic extension sqrt(2).

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

Marc LeBrun, Dec 17 2003

Analog of Moebius mu with sqrt(2) adjoined. Same as mu (A008683) except: 0 for even n (A005843) due to square (extended prime) factor (sqrt(2))^2 and rational primes of the form 8k+/-1 (A001132) factor into conjugate (extended prime) pairs (a + b*sqrt(2))(a - b*sqrt(2)), thus contributing +1 to the product instead of -1; e.g., 7 = (3 + sqrt(2))(3 - sqrt(2)).

For even n a(n) must be 0 because 2 is a square in the quadratic field and so the mu-analog is 0. Of course this coincidentally matches the 0's at even n in A087003. For odd n, from its definition as a product, |a(n)| MUST be the same as that of |mu(n)|. Since from the above we know that A087003(n) is the same as mu(n) at odd n, we can conclude that |a(n)| = |A087003(n)| for all n.

			a(21) = (-1)*(+1) = -1 because 21 = 3*7 where 3 and 7 are congruent to +3 and -1 mod 8 respectively.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 256, p. 221.

Antti Karttunen, Table of n, a(n) for n = 1..20000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Absolute values are the same as those of A087003.
Cf. A008683 (original Moebius function over the integers), A318608 (Moebius function over Z[sqrt(i)], also having the same absolute value as a(n)).
Cf. A001132, A005843, A091337.

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,8)==3||Mod(p,8)==5)&e==1, r*=-1);
    );
    return(r);
} \\ Jianing Song, Aug 30 2018

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - |p mod 8|) where the product is taken over the primes.
From Jianing Song, Aug 30 2018: (Start)
Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == +-1 (mod 8) and -1 if p == +-3 (mod 8).
For squarefree n, a(n) = Kronecker symbol (n, 2) (or (2, n)) = A091337(n). Also for these n, a(n) = A318608(n) if n even or n == 1 (mod 8), otherwise -A318608(n).
(End)

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