A319449 Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.

1, 5, 13, 21, 26, 65, 64, 85, 121, 130, 122, 273, 196, 320, 338, 341, 290, 605, 400, 546, 832, 610, 530, 1105, 651, 980, 1093, 1344, 842, 1690, 1024, 1365, 1586, 1450, 1664, 2541, 1444, 2000, 2548, 2210, 1682, 4160, 1936, 2562, 3146, 2650, 2210, 4433, 3249, 3255

Offset

1,2

Comments

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

Links

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

Wikipedia, Eisenstein integer

Formula

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

Example

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2, and ||d|| denote the norm of d.
a(3) = ||1|| + ||1 + w|| + ||3|| = 1 + 3 + 9 = 13.
a(7) = ||1|| + ||2 + w|| + ||2 + w'|| + ||7|| = 1 + 7 + 7 + 49 = 64.

Mathematica

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

Prog

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

Crossrefs

Cf. A001157.

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), this sequence ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).

Equivalent in the ring of Gaussian integers: A317797.

Sequence in context: A190372 A184837 A197120 * A030374 A369021 A007675

Adjacent sequences: A319446 A319447 A319448 * A319450 A319451 A319452

Keyword

nonn,mult

Author

Jianing Song, Sep 19 2018

Status

approved