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A318608 Moebius function mu(n) defined for the Gaussian integers. 5
1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Just like the original Moebius function over the integers, a(n) = 0 if n has a squared Gaussian prime factor, otherwise (-1)^t if n is a product of a Gaussian unit and t distinct Gaussian prime factors.

a(n) = 0 for even n since 2 = -i*(1 + i)^2 contains a squared factor. For rational primes p == 1 (mod 4), p is always factored as (x + y*i)(x - y*i), x + y*i and x - y*i are not associated so a(p) = (-1)*(-1) = 1.

Interestingly, a(n) and A091069(n) have the same absolute value (= |A087003(n)|), since the discriminants of the quadratic fields Q[i] and Q[sqrt(2)] are -4 and 8 respectively, resulting in Q[i] and Q[sqrt(2)] being two of the three quadratic fields with discriminant a power of 2 or negated (the other one being Q[sqrt(-2)] with discriminant -8).

LINKS

Jianing Song, Table of n, a(n) for n = 1..10000

Wikipedia, Gaussian integer

FORMULA

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - (p mod 4)) where the product is taken over the primes.

Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == 1 (mod 4) and -1 if p == 3 (mod 4).

a(n) = 0 if A078458(n) != A086275(n), otherwise (-1)^A086275(n).

a(n) = A103226(n,0) = A103226(0,n).

For squarefree n, a(n) = Kronecker symbol (-4, n) = A101455(n). Also for these n, a(n) = A091069(n) if n even or n == 1 (mod 8), otherwise -A091069(n).

EXAMPLE

a(15) = -1 because 15 is factored as 3*(2 + i)*(2 - i) with three distinct Gaussian prime factors.

a(21) = (-1)*(-1) = 1 because 21 = 3*7 where 3 and 7 are congruent to 3 mod 4 (thus being Gaussian primes).

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||e>=2, r=0);

        if(Mod(p, 4)==3&e==1, r*=-1);

    );

    return(r);

}

CROSSREFS

Absolute values are the same as those of A087003.

First row and column of A103226.

Cf. A008683 (original Moebius function over the integers), A091069 (Moebius function over Z[sqrt(2)]).

Cf. A101455.

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), A317797 ("sigma", A000203), A079458 ("phi", A000010), A227334 ("psi", A002322), A086275 ("omega", A001221), A078458 ("Omega", A001222), this sequence ("mu", A008683).

Sequence in context: A080545 A099991 A091069 * A087003 A104606 A014389

Adjacent sequences:  A318605 A318606 A318607 * A318609 A318610 A318611

KEYWORD

sign,easy,mult

AUTHOR

Jianing Song, Aug 30 2018

STATUS

approved

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Last modified December 10 23:33 EST 2018. Contains 318049 sequences. (Running on oeis4.)