

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
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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 muanalog 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==2e>=2, r=0);
if((Mod(p, 8)==3Mod(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



