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A062327 Number of divisors of n over the Gaussian integers.

%I A062327 #40 Dec 06 2024 10:24:46
%S A062327 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 A062327 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 A062327 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 A062327 Number of divisors of n over the Gaussian integers.
%C A062327 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 A062327 a(A004614(n)) = A000005(n). - _Vladeta Jovovic_, Jan 23 2003
%C A062327 a(A004613(n)) = A000005(n)^2. - _Benedikt Otten_, May 22 2013
%H A062327 T. D. Noe, <a href="/A062327/b062327.txt">Table of n, a(n) for n = 1..10000</a>
%H A062327 <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%F A062327 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 A062327 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 A062327 For example, 5 has divisors 1, 1+2i, 2+i and 5.
%p A062327 a:= n-> mul(`if`(i[1]=2, 2*i[2]+1, `if`(irem(i[1], 4)=3,
%p A062327                  i[2]+1, (i[2]+1)^2)), i=ifactors(n)[2]):
%p A062327 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jul 09 2021
%t A062327 Table[Length[Divisors[n, GaussianIntegers -> True]], {n, 30}] (* _Alonso del Arte_, Jan 25 2011 *)
%t A062327 DivisorSigma[0,Range[90],GaussianIntegers->True] (* _Harvey P. Dale_, Mar 19 2017 *)
%o A062327 (Haskell)
%o A062327 a062327 n = product $ zipWith f (a027748_row n) (a124010_row n) where
%o A062327    f 2 e                  = 2 * e + 1
%o A062327    f p e | p `mod` 4 == 1 = (e + 1) ^ 2
%o A062327          | otherwise      = e + 1
%o A062327 -- _Reinhard Zumkeller_, Oct 18 2011
%o A062327 (PARI)
%o A062327 a(n)=
%o A062327 {
%o A062327     my(r=1,f=factor(n));
%o A062327     for(j=1,#f[,1], my(p=f[j,1],e=f[j,2]);
%o A062327         if(p==2,r*=(2*e+1));
%o A062327         if(p%4==1,r*=(e+1)^2);
%o A062327         if(p%4==3,r*=(e+1));
%o A062327     );
%o A062327     return(r);
%o A062327 }  \\ _Joerg Arndt_, Dec 09 2016
%Y A062327 Cf. A027748, A124010.
%Y A062327 Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): this sequence ("d", A000005), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
%Y A062327 Equivalent in the ring of Eisenstein integers: A319442.
%K A062327 nonn,nice,mult
%O A062327 1,2
%A A062327 _Reiner Martin_, Jul 12 2001