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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
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)).
Sequence in context: A080545 A355820 A099991 * A318608 A087003 A266840
KEYWORD
mult,easy,sign
AUTHOR
Marc LeBrun, Dec 17 2003
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)