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A319442 Number of divisors of n over the Eisenstein integers.

%I A319442 #27 Dec 14 2024 09:16:10
%S A319442 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,
%T A319442 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,
%U A319442 8,20,7,8,12,4,6,6,16,2,20,4,8,9,12,8,24,4,10
%N A319442 Number of divisors of n over the Eisenstein integers.
%C A319442 Equivalent of d (A000005) in the ring of Eisenstein integers.
%C A319442 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).
%H A319442 Jianing Song, <a href="/A319442/b319442.txt">Table of n, a(n) for n = 1..10000</a>
%H A319442 Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%F A319442 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).
%e A319442 Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.
%e A319442 Divisors of 7 over the Eisenstein integers are 1, 2 + w, 2 + w', 7 and their association, so a(7) = 4.
%e A319442 Divisors of 9 over the Eisenstein integers are 1, 1 + w, 3, 3 + 3w, 9 and their association, so a(9) = 5.
%p A319442 A319442 := proc(n) local t, f, j, e, m; t := 1: f := ifactors(n)[2];
%p A319442    for j from 1 to nops(f) do
%p A319442       e := f[j, 2] + 1; m := f[j, 1] mod 3;
%p A319442       if   m = 0 then 2*e-1
%p A319442       elif m = 1 then e^2
%p A319442       else e fi;
%p A319442       t := t * % od;
%p A319442 t end: seq(A319442(n), n=1..80); # _Peter Luschny_, Oct 03 2018
%t A319442 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 *)
%o A319442 (PARI)
%o A319442 A319442(n)=
%o A319442 {
%o A319442     my(r=1, f=factor(n));
%o A319442     for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
%o A319442         if(p==3, r*=(2*e+1));
%o A319442         if(p%3==1, r*=(e+1)^2);
%o A319442         if(p%3==2, r*=(e+1));
%o A319442     );
%o A319442     return(r);
%o A319442 }
%Y A319442 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).
%Y A319442 Equivalent in the ring of Gaussian integers: A062327.
%K A319442 nonn,mult
%O A319442 1,2
%A A319442 _Jianing Song_, Sep 19 2018