|
%I #25 Feb 12 2020 06:09:10
%S 1,7,10,31,36,70,50,127,91,252,122,310,196,350,360,511,324,637,362,
%T 1116,500,854,530,1270,961,1372,820,1550,900,2520,962,2047,1220,2268,
%U 1800,2821,1444,2534,1960,4572,1764,3500,1850,3782,3276,3710,2210,5110,2451,6727
%N Sum of the norm of divisors of n over Gaussian integers, with associated divisors counted only once.
%C Equivalent of sigma (A000203) in the ring of Gaussian integers. Note that only norms are summed up.
%H Jianing Song, <a href="/A317797/b317797.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gaussian_integer">Gaussian integer</a>
%F 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).
%e Let ||d|| denote the norm of d.
%e a(2) = ||1|| + ||1 + i|| + ||2|| = 1 + 2 + 4 = 7.
%e 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.
%t 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 *)
%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^(2*e+1)-1));
%o if(Mod(p,4)==1, r*=((p^(e+1)-1)/(p-1))^2);
%o if(Mod(p,4)==3, r*=(p^(2*e+2)-1)/(p^2-1));
%o );
%o return(r);
%o }
%Y Cf. A001157.
%Y 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).
%K nonn,mult,look
%O 1,2
%A _Jianing Song_, Aug 07 2018
|
