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.
KEYWORD
nonn,mult
AUTHOR
Jianing Song, Sep 19 2018
STATUS
approved