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Search: id:A061446
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| A061446 |
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Primitive part of Fibonacci(n). |
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+0
14
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| 1, 1, 2, 3, 5, 4, 13, 7, 17, 11, 89, 6, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 46, 15005, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 321, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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Fib(n) = A000045(n) = Prod(a(k),k divides n), Lucas(n) = A000204(n) = Prod(a(k),k divides 2n and 2^m|k iff 2^m|2n) (e.g. Lucas(4) = 7 = a(8), Lucas(6) = 18 = a(12)*a(4)) - Len Smiley (smiley(AT)math.uaa.alaska.edu), Nov 11 2001
A 2001 Iranian Mathematical Olympiad question shows such a sequence exists whenever gcd(a(m),a(n)) = a(gcd(m,n)).
The problem of characterization family of all GCD-morphic sequences F i.e. such that GCD(F(m),F(n)) = F(GCD(m,n)) was posed by A.K.Kwasniewski (GCD-morphic Problem). Dziemianczuk and Bajguz (2008) showed that any GCD-morphic sequence is coded by certain natural number-valued sequence. [From M. Dziemianczuk (maciek.ciupa(AT)gmail.com), Jan 15 2009]
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REFERENCES
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Brillhart, John; Montgomery, Peter L.; Silverman, Robert D.; Tables of Fibonacci and Lucas factorizations. Math. Comp. 50 (1988), no. 181, 251-260, S1-S15. Math. Rev. 89h:11002.
R. D. Carmichael, On the Numerical Factors of the Arithmetic Forms (alpha)^n + (beta)^n, Ann. of Math. 15, (1913), pp. 30-48.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
C. K. Caldwell, Lucas Aurifeuillian primitive part
M. Dziemianczuk and W. Bajguz, On GCD-morphic sequences, ArXiv:0802.1303 [From M. Dziemianczuk (maciek.ciupa(AT)gmail.com), Jan 15 2009]
A. K. Kwasniewski, Cobweb posets as noncommutative prefabs, Adv. Stud. Contemp. Math. vol.14 (1) 2007. pp. 37-47 [From M. Dziemianczuk (maciek.ciupa(AT)gmail.com), Jan 15 2009]
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FORMULA
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Let r=(1+sqrt(5))/2. For n>2, the primitive part of F(n)=(r^n-(-1/r)^n)/sqrt(5) is Phi_n(-r^2)/r^phi(n) where Phi_n is n-th cyclotomic polynomial and phi is Euler's totient function A000010.
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CROSSREFS
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Cf. A061447, A061254, A061445, A061442, A061443.
Cf. A105602, A126025, A126069.
Sequence in context: A023395 A101409 A131401 this_sequence A107476 A094140 A119745
Adjacent sequences: A061443 A061444 A061445 this_sequence A061447 A061448 A061449
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KEYWORD
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nonn,new
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AUTHOR
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D.Broadhurst(AT)open.ac.uk, Jun 10 2001
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EXTENSIONS
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More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 09 2001
Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Plewe, May 29 2007
Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 28 2009
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