OFFSET
1,2
COMMENTS
Equivalent of sigma (A000203) in the ring of Gaussian integers. Note that only norms are summed up.
LINKS
Jianing Song, Table of n, a(n) for n = 1..10000
Wikipedia, Gaussian integer
FORMULA
Multiplicative with a(2^e) = sigma(2^(2e)) = 2^(2e+1) - 1, a(p^e) = sigma(p^e)^2 = ((p^(e+1) - 1)/(p - 1))^2 if p == 1 (mod 4) and sigma_2(p^e) = A001157(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 3 (mod 4).
EXAMPLE
Let ||d|| denote the norm of d.
a(2) = ||1|| + ||1 + i|| + ||2|| = 1 + 2 + 4 = 7.
a(5) = ||1|| + ||2 + i|| + ||2 - i|| + ||5|| = 1 + 5 + 5 + 25 = 36. Note that 2 - i and 1 + 2i are associated so their norm is only counted once.
MATHEMATICA
f[p_, e_] := If[p == 2, 2^(2*e + 1) - 1, Switch[Mod[p, 4], 1, ((p^(e + 1) - 1)/(p - 1))^2, 3, (p^(2 e + 2) - 1)/(p^2 - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Feb 12 2020 *)
PROG
(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^(2*e+1)-1));
if(Mod(p, 4)==1, r*=((p^(e+1)-1)/(p-1))^2);
if(Mod(p, 4)==3, r*=(p^(2*e+2)-1)/(p^2-1));
);
return(r);
}
CROSSREFS
Cf. A001157.
Equivalent of arithmetic functions in the ring of Gaussian integers (the corresponding functions in the ring of integers are in the parentheses): A062327 ("d", A000005), this sequence ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), A318608 ("mu", A008683).
KEYWORD
AUTHOR
Jianing Song, Aug 07 2018
STATUS
approved