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A091069 Moebius mu sequence for real quadratic extension sqrt(2). 3
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

Analog of Moebius mu with sqrt(2) adjoined. Same as mu (A008683) except: 0 for even n (A005843) due to square (extended prime) factor (sqrt(2))^2 and rational primes of the form 8k+/-1 (A001132) factor into conjugate (extended prime) pairs (a + b*sqrt(2))(a - b*sqrt(2)), thus contributing +1 to the product instead of -1; e.g., 7 = (3 + sqrt(2))(3 - sqrt(2)).

For even n a(n) must be 0 because 2 is a square in the quadratic field and so the mu-analog is 0. Of course this coincidentally matches the 0's at even n in A087003. For odd n, from its definition as a product, |a(n)| MUST be the same as that of |mu(n)|. Since from the above we know that A087003(n) is the same as mu(n) at odd n, we can conclude that |a(n)| = |A087003(n)| for all n.

REFERENCES

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 256, p. 221.

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..20000

Antti Karttunen, Data supplement: n, a(n) computed for n =  1..100000

FORMULA

a(n) = 0 if n even or has a square prime factor, otherwise Product_{p divides n} (2 - |p mod 8|) where the product is taken over the primes.

From Jianing Song, Aug 30 2018: (Start)

Multiplicative with a(p^e) = 0 if p = 2 or e > 1, a(p) = 1 if p == +-1 (mod 8) and -1 if p == +-3 (mod 8).

For squarefree n, a(n) = Kronecker symbol (n, 2) (or (2, n)) = A091337(n). Also for these n, a(n) = A318608(n) if n even or n == 1 (mod 8), otherwise -A318608(n).

(End)

EXAMPLE

a(21) = (-1)*(+1) = -1 because 21 = 3*7 where 3 and 7 are congruent to +3 and -1 mod 8 respectively.

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, 8)==3||Mod(p, 8)==5)&e==1, r*=-1);

    );

    return(r);

} \\ Jianing Song, Aug 30 2018

CROSSREFS

Absolute values are the same as those of A087003.

Cf. A008683 (original Moebius function over the integers), A318608 (Moebius function over Z[sqrt(i)], also having the same absolute value as a(n)).

Cf. A001132, A005843, A091337.

Sequence in context: A285128 A080545 A099991 * A318608 A087003 A266840

Adjacent sequences:  A091066 A091067 A091068 * A091070 A091071 A091072

KEYWORD

mult,easy,sign

AUTHOR

Marc LeBrun, Dec 17 2003

STATUS

approved

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Last modified March 18 17:51 EDT 2019. Contains 321292 sequences. (Running on oeis4.)