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A317797 Sum of the norm of divisors of n over Gaussian integers, with associated divisors counted only once. 6
1, 7, 10, 31, 36, 70, 50, 127, 91, 252, 122, 310, 196, 350, 360, 511, 324, 637, 362, 1116, 500, 854, 530, 1270, 961, 1372, 820, 1550, 900, 2520, 962, 2047, 1220, 2268, 1800, 2821, 1444, 2534, 1960, 4572, 1764, 3500, 1850, 3782, 3276, 3710, 2210, 5110, 2451, 6727 (list; graph; refs; listen; history; text; internal format)
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).

Sequence in context: A174466 A070422 A102574 * A284418 A244165 A119169

Adjacent sequences:  A317794 A317795 A317796 * A317798 A317799 A317800

KEYWORD

nonn,mult,look

AUTHOR

Jianing Song, Aug 07 2018

STATUS

approved

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Last modified September 26 05:00 EDT 2022. Contains 356986 sequences. (Running on oeis4.)