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A319448 Moebius function mu(n) defined for the Eisenstein integers. 7
1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 0, 1, -1, 0, 1, -1, 0, 0, -1, 0, 1, 0, 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 Eisenstein prime factor, otherwise (-1)^t if n is a product of an Eisenstein unit and t distinct Eisenstein prime factors.

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2. a(n) = 0 for n divisible by 3 since 3 = w'*(1 + w)^2 contains a squared factor. For rational primes p == 1 (mod 3), p is always factored as (x + y*w)(x + y*w'), x + y*w and x + y*w' are not associated so a(p) = (-1)*(-1) = 1.

LINKS

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

Wikipedia, Eisenstein integer

FORMULA

a(n) = 0 if n is divisible by 3 or has a square prime factor, otherwise Product_{p divides n} (3 - 2*(p mod 3)) where the product is taken over the primes.

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

For squarefree n, a(n) = Legendre symbol (n, 3) = Kronecker symbol (-3, n) = A102283(n).

EXAMPLE

Let w = (1 + sqrt(3)*i)/2, w' = (1 - sqrt(3)*i)/2.

a(14) = -1 because 14 is factored as 2*(2 + w)*(2 + w') with three distinct Eisenstein prime factors.

a(55) = (-1)*(-1) = 1 because 55 = 5*11 where 5 and 11 are congruent to 2 mod 3 (thus being Eisenstein 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==3||e>=2, r=0);

        if(Mod(p, 3)==2&e==1, r*=-1);

    );

    return(r);

}

CROSSREFS

Cf. A102283.

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), A319449 ("sigma", A000203), A319445 ("phi", A000010), A319446 ("psi", A002322), A319443 ("omega", A001221), A319444 ("Omega", A001222), this sequence ("mu", A008683).

Equivalent in the ring of Gaussian integers: A318608.

Sequence in context: A074711 A004585 A156277 * A260595 A177444 A239200

Adjacent sequences:  A319445 A319446 A319447 * A319449 A319450 A319451

KEYWORD

sign,mult

AUTHOR

Jianing Song, Sep 19 2018

STATUS

approved

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Last modified January 21 19:57 EST 2019. Contains 319350 sequences. (Running on oeis4.)