

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==2e>=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



