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%I #21 Feb 10 2020 18:06:33
%S 1,5,13,21,26,65,64,85,121,130,122,273,196,320,338,341,290,605,400,
%T 546,832,610,530,1105,651,980,1093,1344,842,1690,1024,1365,1586,1450,
%U 1664,2541,1444,2000,2548,2210,1682,4160,1936,2562,3146,2650,2210,4433,3249,3255
%N Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.
%C Equivalent of sigma (A000203) in the ring of Eisenstein integers. Note that only norms are summed up.
%H Jianing Song, <a href="/A319449/b319449.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Eisenstein_integer">Eisenstein integer</a>
%F Multiplicative with a(3^e) = sigma(3^(2e)) = (3^(2e+1) - 1)/2, a(p^e) = sigma(p^e)^2 = ((p^(e+1) - 1)/(p - 1))^2 if p == 1 (mod 3) and sigma_2(p^e) = A001157(p^e) = (p^(2e+2) - 1)/(p^2 - 1) if p == 2 (mod 3).
%e Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2, and ||d|| denote the norm of d.
%e a(3) = ||1|| + ||1 + w|| + ||3|| = 1 + 3 + 9 = 13.
%e a(7) = ||1|| + ||2 + w|| + ||2 + w'|| + ||7|| = 1 + 7 + 7 + 49 = 64.
%t f[p_, e_] := If[p == 3 , DivisorSigma[1, 3^(2*e)], Switch[Mod[p, 3], 1, DivisorSigma[1, p^e]^2, 2, DivisorSigma[2, p^e]]]; eisSigma[1] = 1; eisSigma[n_] := Times @@ f @@@ FactorInteger[n]; Array[eisSigma, 100] (* _Amiram Eldar_, Feb 10 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==3, r*=((3^(2*e+1)-1)/2));
%o if(Mod(p, 3)==1, r*=((p^(e+1)-1)/(p-1))^2);
%o if(Mod(p, 3)==2, 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 Eisenstein integers (the corresponding functions in the ring of integers are in the parentheses): A319442 ("d", A000005), this sequence ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), A319448 ("mu", A008683).
%Y Equivalent in the ring of Gaussian integers: A317797.
%K nonn,mult
%O 1,2
%A _Jianing Song_, Sep 19 2018
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