|
%I #29 Nov 14 2018 08:11:24
%S 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,
%T 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,
%U 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
%N Moebius mu sequence for real quadratic extension sqrt(2).
%C 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)).
%C 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.
%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 256, p. 221.
%H Antti Karttunen, <a href="/A091069/b091069.txt">Table of n, a(n) for n = 1..20000</a>
%H Antti Karttunen, <a href="/A091069/a091069.txt">Data supplement: n, a(n) computed for n = 1..100000</a>
%F 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.
%F From _Jianing Song_, Aug 30 2018: (Start)
%F 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).
%F 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).
%F (End)
%e a(21) = (-1)*(+1) = -1 because 21 = 3*7 where 3 and 7 are congruent to +3 and -1 mod 8 respectively.
%o (PARI)
%o a(n)=
%o {
%o my(r=1, f=factor(n));
%o for(j=1, #f[, 1], my(p=f[j, 1], e=f[j, 2]);
%o if(p==2||e>=2, r=0);
%o if((Mod(p,8)==3||Mod(p,8)==5)&e==1, r*=-1);
%o );
%o return(r);
%o } \\ _Jianing Song_, Aug 30 2018
%Y Absolute values are the same as those of A087003.
%Y Cf. A008683 (original Moebius function over the integers), A318608 (Moebius function over Z[sqrt(i)], also having the same absolute value as a(n)).
%Y Cf. A001132, A005843, A091337.
%K mult,easy,sign
%O 1,1
%A _Marc LeBrun_, Dec 17 2003
|
