A062327 -id:A062327 - cp's OEIS frontend

A332317 Numbers k that are harmonic in Gaussian integers: k * A062327(k) is divisible by A103228(k) + i*A103229(k) (where i is the imaginary unit).

1, 5, 130, 390, 585, 3250, 31980, 133250, 223860, 799500, 7195500, 13591500, 122323500, 258238500, 394153500, 405346500, 910630500, 1345558500, 2025133500, 8195674500

Offset: 1

Amiram Eldar, Feb 09 2020

Analogous to harmonic numbers (A001599), with the number and sum of divisors functions generalized for Gaussian integers (A062327, A103228, A103229) instead of the number and sum of divisors functions (A000005, A000203).

No more terms below 10^10.

			5 is a term since 5 * A062327(5)/(A103228(5) + i*A103229(5)) = 5 * 4 /(4 + 8*i) = 1 - 2*i is a Gaussian integer.

Cf. A000005, A000203, A001599, A062327, A103228, A103229, A103230.

Select[Range[10^4], Divisible[# * DivisorSigma[0, #, GaussianIntegers -> True], DivisorSigma[1, #, GaussianIntegers -> True]] &]

A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

Original entry on oeis.org

1, 5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 125, 137, 145, 149, 157, 169, 173, 181, 185, 193, 197, 205, 221, 229, 233, 241, 257, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421

Offset: 1

N. J. A. Sloane

Also gives solutions z to x^2+y^2=z^4 with gcd(x,y,z)=1 and x,y,z positive. - John Sillcox (johnsillcox(AT)hotmail.com), Feb 20 2004
A065338(a(n)) = 1. - Reinhard Zumkeller, Jul 10 2010
Product_{k=1..A001221(a(n))} A079260(A027748(a(n),k)) = 1. - Reinhard Zumkeller, Jan 07 2013
A062327(a(n)) = A000005(a(n))^2. (These are the only numbers that satisfy this equation.) - Benedikt Otten, May 22 2013
Numbers that are positive integer divisors of 1 + 4*x^2 where x is a positive integer. - Michael Somos, Jul 26 2013
Numbers k such that there is a "knight's move" of Euclidean distance sqrt(k) which allows the whole of the 2D lattice to be reached. For example, a knight which travels 4 units in any direction and then 1 unit at right angles to the first direction moves a distance sqrt(17) for each move. This knight can reach every square of an infinite chessboard.
Also 1/7 of the area of the n-th largest octagon with angles 3*Pi/4, along the perimeter of which there are only 8 nodes of the square lattice - at its vertices. - Alexander M. Domashenko, Feb 21 2024
Sequence closed under multiplication. Odd values of A031396 and their powers. These are the only numbers m that satisfy the Pell equation (k*x)^2 - D*(m*y)^2 = -1. - Klaus Purath, May 12 2025

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.

T. D. Noe, Table of n, a(n) for n = 1..10000
David Broadhurst and Wadim Zudilin, A magnetic double integral, arXiv:1708.02381 [math.NT], 2017. See p. 7
Code Golf Stack Exchange, pseudonymous Stack Exchange user 'trichoplax', N-movers: How much of the infinite board can I reach?

Subsequence of A000404; A002144 is a subsequence. Essentially same as A008846.

Cf. A004614.

a004613 n = a004613_list !! (n-1)
a004613_list = filter (all (== 1) . map a079260 . a027748_row) [1..]
-- Reinhard Zumkeller, Jan 07 2013

[n: n in [1..500] | forall{d: d in PrimeDivisors(n) | d mod 4 eq 1}]; // Vincenzo Librandi, Aug 21 2012

isA004613 := proc(n)
    local p;
    for p in numtheory[factorset](n) do
        if modp(p,4) <> 1 then
            return false;
        end if;
    end do:
    true;
end proc:
for n from 1 to 200 do
    if isA004613(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Nov 17 2014
# second Maple program:
q:= n-> andmap(i-> irem(i[1], 4)=1, ifactors(n)[2]):
select(q, [$1..500])[];  # Alois P. Heinz, Jan 13 2024

ok[1] = True; ok[n_] := And @@ (Mod[#, 4] == 1 &) /@ FactorInteger[n][[All, 1]]; Select[Range[421], ok] (* Jean-François Alcover, May 05 2011 *)
Select[Range[500],Union[Mod[#,4]&/@(FactorInteger[#][[All,1]])]=={1}&] (* Harvey P. Dale, Mar 08 2017 *)

for(n=1,1000,if(sumdiv(n,d,isprime(d)*if((d-1)%4,1,0))==0,print1(n,",")))

is(n)=n%4==1 && factorback(factor(n)[,1]%4)==1 \\ Charles R Greathouse IV, Sep 19 2016

Numbers of the form x^2 + y^2 where x is even, y is odd and gcd(x, y) = 1.

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

Original entry on oeis.org

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

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

A319442 Number of divisors of n over the Eisenstein integers.

Original entry on oeis.org

1, 2, 3, 3, 2, 6, 4, 4, 5, 4, 2, 9, 4, 8, 6, 5, 2, 10, 4, 6, 12, 4, 2, 12, 3, 8, 7, 12, 2, 12, 4, 6, 6, 4, 8, 15, 4, 8, 12, 8, 2, 24, 4, 6, 10, 4, 2, 15, 9, 6, 6, 12, 2, 14, 4, 16, 12, 4, 2, 18, 4, 8, 20, 7, 8, 12, 4, 6, 6, 16, 2, 20, 4, 8, 9, 12, 8, 24, 4, 10

Offset: 1

Jianing Song, Sep 19 2018

Equivalent of d (A000005) in the ring of Eisenstein integers.

Divisors which are associates are identified (two Eisenstein integers z1, z2 are associates if z1 = u * z2 where u is an Eisenstein unit, i.e., one of +-1 or (+-1 +- sqrt(3)*i)/2).

			Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
Divisors of 7 over the Eisenstein integers are 1, 2 + w, 2 + w', 7 and their association, so a(7) = 4.
Divisors of 9 over the Eisenstein integers are 1, 1 + w, 3, 3 + 3w, 9 and their association, so a(9) = 5.

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

Equivalent of arithmetic functions in the ring of Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A062327.

A319442 := proc(n) local t, f, j, e, m; t := 1: f := ifactors(n)[2];
   for j from 1 to nops(f) do
      e := f[j, 2] + 1; m := f[j, 1] mod 3;
      if   m = 0 then 2*e-1
      elif m = 1 then e^2
      else e fi;
      t := t * % od;
t end: seq(A319442(n), n=1..80); # Peter Luschny, Oct 03 2018

f[p_, e_] := Switch[Mod[p, 3], 0, 2*e + 1, 1, (e + 1)^2, 2, e + 1]; eisNumDiv[1] = 1; eisNumDiv[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisNumDiv, 100] (* Amiram Eldar, Feb 10 2020 *)

A319442(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*=(2*e+1));
        if(p%3==1, r*=(e+1)^2);
        if(p%3==2, r*=(e+1));
    );
    return(r);
}

Multiplicative with a(3^e) = 2*e + 1, a(p^e) = (e + 1)^2 if p == 1 (mod 3) and e + 1 if p == 2 (mod 3).

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).

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).

A078910 Let r+i*s be the sum of the distinct first-quadrant Gaussian integers dividing n; sequence gives r values.

Original entry on oeis.org

1, 4, 4, 10, 9, 16, 8, 22, 13, 37, 12, 40, 19, 32, 36, 46, 23, 52, 20, 93, 32, 48, 24, 88, 56, 77, 40, 80, 37, 148, 32, 94, 48, 95, 72, 130, 45, 80, 76, 205, 51, 128, 44, 120, 117, 96, 48, 184, 57, 231, 92, 193, 63, 160, 108, 176, 80, 151, 60, 372, 73, 128, 104, 190, 176

Offset: 1

N. J. A. Sloane, Jan 11 2003

A Gaussian integer z = x+iy is in the first quadrant if x > 0, y >= 0. Just one of the 4 associates z, -z, i*z, -i*z is in the first quadrant.

			The distinct first-quadrant divisors of 4 are 1, 1+i, 2, 2+2*i, 4, with sum 10+3*i, so a(4) = 10.

M. F. Hasler, Table of n, a(n) for n = 1..1000.
Michael Somos, PARI program for finding prime decomposition of Gaussian integers
Index entries for Gaussian integers and primes

Cf. A000203, A078911, A078458, A078908, A078909, A078930.

Cf. A062327 for the number of first quadrant divisors of n.

Table[Re[Plus@@Divisors[n, GaussianIntegers -> True]], {n, 65}] (* Alonso del Arte, Jan 24 2012 *)

A078910(n,S=[])=sigma(n)+sumdiv(n*I,d,if(real(d)&imag(d)&!setsearch(S,d=vecsort(abs([real(d),imag(d)]))),S=setunion(S,[d]);(d[1]+d[2])>>(d[1]==d[2]))) \\ M. F. Hasler, Nov 22 2007

a(n) = A078911(n)+A000203(n). - Vladeta Jovovic, Jan 11 2003

More terms from Vladeta Jovovic, Jan 11 2003

cp's OEIS Frontend

A332317 Numbers k that are harmonic in Gaussian integers: k * A062327(k) is divisible by A103228(k) + i*A103229(k) (where i is the imaginary unit).

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A004613 Numbers that are divisible only by primes congruent to 1 mod 4.

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

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

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A319442 Number of divisors of n over the Eisenstein integers.

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

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

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