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A319449
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Sum of the norm of divisors of n over Eisenstein integers, with associated divisors counted only once.
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7
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1, 5, 13, 21, 26, 65, 64, 85, 121, 130, 122, 273, 196, 320, 338, 341, 290, 605, 400, 546, 832, 610, 530, 1105, 651, 980, 1093, 1344, 842, 1690, 1024, 1365, 1586, 1450, 1664, 2541, 1444, 2000, 2548, 2210, 1682, 4160, 1936, 2562, 3146, 2650, 2210, 4433, 3249, 3255
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OFFSET
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1,2
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COMMENTS
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Equivalent of sigma (A000203) in the ring of Eisenstein integers. Note that only norms are summed up.
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LINKS
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FORMULA
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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).
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EXAMPLE
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Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2, and ||d|| denote the norm of d.
a(3) = ||1|| + ||1 + w|| + ||3|| = 1 + 3 + 9 = 13.
a(7) = ||1|| + ||2 + w|| + ||2 + w'|| + ||7|| = 1 + 7 + 7 + 49 = 64.
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MATHEMATICA
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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 *)
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PROG
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(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==3, r*=((3^(2*e+1)-1)/2));
if(Mod(p, 3)==1, r*=((p^(e+1)-1)/(p-1))^2);
if(Mod(p, 3)==2, r*=(p^(2*e+2)-1)/(p^2-1));
);
return(r);
}
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CROSSREFS
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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).
Equivalent in the ring of Gaussian integers: A317797.
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KEYWORD
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nonn,mult
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AUTHOR
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STATUS
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approved
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