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A154269
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Dirichlet inverse of A019590; Fully multiplicative with a(2^e) = (-1)^e, a(p^e) = 0 for odd primes p.
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15
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1, -1, 0, 1, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET
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1,1
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COMMENTS
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Equals +1 if n is an even power of 2 (2^0, 2^2, 2^4,...), -1 if n is an odd power of 2 (2^1, 2^3, 2^5,..) and zero anywhere else.
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LINKS
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FORMULA
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a(n) is multiplicative with a(2^e) = (-1)^e, a(p^e) = 0^e if p>2. - Michael Somos, Jul 05 2009
G.f. A(x) satisfies x = A(x) + A(x^2).
a(1) = 1, after which: a(2n) = -a(n), a(2n+1) = 0. - Antti Karttunen, Jul 24 2017
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EXAMPLE
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x - x^2 + x^4 - x^8 + x^16 - x^32 + x^64 - x^128 + x^256 - x^512 + ...
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MAPLE
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a:= n-> (p-> `if`(2^p=n, (-1)^p, 0))(ilog2(n)):
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MATHEMATICA
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nn = 95; a = PadRight[{1, 1}, nn, 0]; Inverse[Table[Table[If[Mod[n, k] == 0, a[[n/k]], 0], {k, 1, nn}], {n, 1, nn}]][[All, 1]] (* Mats Granvik, Jul 24 2017 *)
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PROG
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(PARI) {a(n) = if( n < 2, n == 1, - a(n / 2))} /* Michael Somos, Jul 05 2009 */
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CROSSREFS
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Cf. A209229 (gives the absolute values).
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KEYWORD
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sign,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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