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

Author

Jianing Song, Sep 19 2018

Keywords

• nonn
• mult

Comments

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

Examples

			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.
		

Links

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

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.

Programs

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 *)

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);
}

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).