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%I #36 Oct 30 2022 18:19:59
%S 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,
%T 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,
%U 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
%N Number of divisors of n over the Gaussian integers.
%C 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).
%C a(A004614(n)) = A000005(n). - _Vladeta Jovovic_, Jan 23 2003
%C a(A004613(n)) = A000005(n)^2. - _Benedikt Otten_, May 22 2013
%H T. D. Noe, <a href="/A062327/b062327.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%F 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
%F 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
%e For example, 5 has divisors 1, 1+2i, 2+i and 5.
%p a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
%p i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jul 09 2021
%t Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* _Alonso del Arte_, Jan 25 2011 *)
%t DivisorSigma[0,Range[90],GaussianIntegers->True] (* _Harvey P. Dale_, Mar 19 2017 *)
%o (Haskell)
%o a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
%o f 2 e = 2 * e + 1
%o f p e | p `mod` 4 == 1 = (e + 1) ^ 2
%o | otherwise = e + 1
%o -- _Reinhard Zumkeller_, Oct 18 2011
%o (PARI)
%o a(n)=
%o {
%o my(r=1,f=factor(n));
%o for(j=1,#f[,1], my(p=f[j,1],e=f[j,2]);
%o if(p==2,r*=(2*e+1));
%o if(p%4==1,r*=(e+1)^2);
%o if(p%4==3,r*=(e+1));
%o );
%o return(r);
%o } \\ _Joerg Arndt_, Dec 09 2016
%Y Cf. A027748, A124010.
%K nonn,nice,mult
%O 1,2
%A _Reiner Martin_, Jul 12 2001
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