A319442 Number of divisors of n over the Eisenstein integers.

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

Author

Jianing Song, Sep 19 2018

Keywords

• nonn
• mult

Comments

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).

Examples

			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.
		

Links

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

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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);
}

Formula

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).