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

1,2

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.

Links

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

Wikipedia, Eisenstein integer

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).

Example

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.

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 *)

Prog

(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);
}

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.

Sequence in context: A345287 A023139 A328484 * A299772 A304311 A175393

Adjacent sequences: A319439 A319440 A319441 * A319443 A319444 A319445

Keyword

nonn,mult

Author

Jianing Song, Sep 19 2018

Status

approved