%I A091069
%S A091069 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,
%T A091069 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,
%U A091069 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
%V A091069 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,
%W A091069 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,
%X A091069 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 A091069 Moebius mu sequence for real quadratic extension sqrt(2).
%C A091069 Analog of Moebius mu with sqrt(2) adjoined. Same as mu (A008683) except: 
               0 for even n (A005843) due to square (extended prime) factor (sqrt2)^2 
               and rational primes of the form 8k+/-1 (A001132) factor into conjugate 
               (extended prime) pairs (a + b sqrt2)(a - b sqrt2), thus contributing 
               +1 to the product instead of -1; e.g. 7 = (3+sqrt2)(3-sqrt2).
%C A091069 First, for even n a(n) must be 0 because 2 is a square in the quadratic 
               field and so the mu-analogue is 0. Of course this coincidentally 
               matches the 0's at even n in A087003. Further, from its definition 
               as a product, |a(n)| MUST be the same as that of |mu|. Since from 
               the above we know that A087003 is the same as mu at odd n, we can 
               conclude that |a(n)| = |A087003| for all n.
%D A091069 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 
               5th ed., Oxford Univ. Press, 1979, Theorem 256, p. 221.
%F A091069 Zero if n even or has a square prime factor, otherwise product 2-|p mod 
               8| for each prime p dividing n (i.e. +1 if p=8k+/-1, -1 if p=8k+/
               -3).
%e A091069 a(21) = (-1)*(+1) = -1 because 21=3*7 which are respectively +3 and -1 
               mod 8
%Y A091069 Absolute values are the same as those of A087003.
%Y A091069 Cf. A008683, A005843, A001132.
%Y A091069 Sequence in context: A100060 A147850 A099991 this_sequence A087003 A104606 
               A014389
%Y A091069 Adjacent sequences: A091066 A091067 A091068 this_sequence A091070 A091071 
               A091072
%K A091069 mult,easy,sign
%O A091069 1,1
%A A091069 Marc LeBrun (mlb(AT)well.com), Dec 17 2003