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A062327 Number of divisors of n over the Gaussian integers. 6
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 (list; graph; refs; listen; history; text; internal format)
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.

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: A259846 A087669 A053087 * A075491 A089279 A049820

Adjacent sequences:  A062324 A062325 A062326 * A062328 A062329 A062330

KEYWORD

nonn,nice,mult

AUTHOR

Reiner Martin (reinermartin(AT)hotmail.com), Jul 12 2001

STATUS

approved

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Last modified January 17 09:43 EST 2018. Contains 297804 sequences.