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 A319442 Number of divisors of n over the Eisenstein integers. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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);    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 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. Sequence in context: A291300 A023139 A328484 * A299772 A304311 A175393 Adjacent sequences:  A319439 A319440 A319441 * A319443 A319444 A319445 KEYWORD nonn,mult AUTHOR Jianing Song, Sep 19 2018 STATUS approved

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Last modified January 26 23:05 EST 2020. Contains 331289 sequences. (Running on oeis4.)