A062327 Number of divisors of n over the Gaussian integers.

1, 3, 2, 5, 4, 6, 2, 7, 3, 12, 2, 10, 4, 6, 8, 9, 4, 9, 2, 20, 4, 6, 2, 14, 9, 12, 4, 10, 4, 24, 2, 11, 4, 12, 8, 15, 4, 6, 8, 28, 4, 12, 2, 10, 12, 6, 2, 18, 3, 27, 8, 20, 4, 12, 8, 14, 4, 12, 2, 40, 4, 6, 6, 13, 16, 12, 2, 20, 4, 24, 2, 21, 4, 12, 18, 10, 4, 24, 2, 36, 5, 12, 2, 20, 16, 6

Offset

1,2

Comments

Divisors which are associates are identified (two Gaussian integers z1, z2 are associates if z1 = u * z2 where u is a unit, i.e., one of 1, i, -1, -i).

a(A004614(n)) = A000005(n). - Vladeta Jovovic, Jan 23 2003

a(A004613(n)) = A000005(n)^2. - Benedikt Otten, May 22 2013

Links

T. D. Noe, Table of n, a(n) for n = 1..10000

Index entries for Gaussian integers and primes

Formula

Presumably a(n) = 2 iff n is a rational prime == 3 mod 4 (see A045326). - N. J. A. Sloane, Jan 07 2003, Feb 23 2007

Multiplicative with a(2^e) = 2*e+1, a(p^e) = e+1 if p mod 4=3 and a(p^e) = (e+1)^2 if p mod 4=1. - Vladeta Jovovic, Jan 23 2003

Example

For example, 5 has divisors 1, 1+2i, 2+i and 5.

Maple

a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
                 i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jul 09 2021

Mathematica

Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* Alonso del Arte, Jan 25 2011 *)
DivisorSigma[0, Range[90], GaussianIntegers->True] (* Harvey P. Dale, Mar 19 2017 *)

Prog

(Haskell)
a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e                  = 2 * e + 1
   f p e | p `mod` 4 == 1 = (e + 1) ^ 2
         | otherwise      = e + 1
-- Reinhard Zumkeller, Oct 18 2011
(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==2, r*=(2*e+1));
        if(p%4==1, r*=(e+1)^2);
        if(p%4==3, r*=(e+1));
    );
    return(r);
}  \\ Joerg Arndt, Dec 09 2016

Crossrefs

Cf. A027748, A124010.

Sequence in context: A087669 A053087 A316519 * A075491 A326730 A089279

Adjacent sequences: A062324 A062325 A062326 * A062328 A062329 A062330

Keyword

nonn,nice,mult

Author

Reiner Martin, Jul 12 2001

Status

approved