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A319442
Number of divisors of n over the Eisenstein integers.
15
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
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)
{
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
KEYWORD
nonn,mult
AUTHOR
Jianing Song, Sep 19 2018
STATUS
approved