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Search: id:A091069
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| A091069 |
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Moebius mu sequence for real quadratic extension sqrt(2). |
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+0
2
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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).
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.
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REFERENCES
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G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Theorem 256, p. 221.
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FORMULA
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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).
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EXAMPLE
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a(21) = (-1)*(+1) = -1 because 21=3*7 which are respectively +3 and -1 mod 8
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CROSSREFS
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Absolute values are the same as those of A087003.
Cf. A008683, A005843, A001132.
Sequence in context: A004569 A100060 A099991 this_sequence A087003 A104606 A014389
Adjacent sequences: A091066 A091067 A091068 this_sequence A091070 A091071 A091072
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KEYWORD
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mult,easy,sign
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AUTHOR
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Marc LeBrun (mlb(AT)well.com), Dec 17 2003
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