A061446 Primitive part of Fibonacci(n).
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
Offset: 1
Keywords
Links
- T. D. Noe, Table of n, a(n) for n = 1..1000
- N. Bliss, B. Fulan, S. Lovett, and J. Sommars, Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials, Amer. Math. Monthly, 120 (2013), 519-536.
- John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), pp. 251-260, S1-S15. Math. Rev. 89h:11002.
- C. K. Caldwell, Lucas Aurifeuillian primitive part
- R. D. Carmichael, On the numerical factors of the arithmetic forms alpha*n+-beta*n, Annals of Math., 2nd ser., 15 (1/4) (1913/14) 30-48.
- Johann Cigler and Hans-Christian Herbig, Factorization of spread polynomials, arXiv:2412.18958 [math.NT], 2024.
- M. Dziemianczuk and W. Bajguz, On GCD-morphic sequences, arXiv:0802.1303 [math.CO], 2008.
- A. K. Kwasniewski, Cobweb posets as noncommutative prefabs, arXiv:math/0503286 [math.CO], 2007; Adv. Stud. Contemp. Math. vol.14 (1) 2007. pp. 37-47.
- Rohit Nagpal and A. Snowden, The module theory of divided power algebras, arXiv preprint arXiv:1606.03431 [math.AC], 2016.
- A. Nowicki, Strong divisibility and LCM-sequences, arXiv:1310.2416 [math.NT], 2013.
- A. Nowicki, Strong divisibility and LCM-sequences, Am. Math. Mnthly 122 (2015), 958-966.
Crossrefs
Programs
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Maple
N:= 200; # to get a(1) to a(N) L[0]:= 1: for i from 1 to N do L[i]:=ilcm(L[i-1],combinat:-fibonacci(i)) od: seq(L[i]/L[i-1],i=1..N); # Robert Israel, Aug 03 2015
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Mathematica
t={1}; Do[f=Fibonacci[n]; Do[f=f/GCD[f,t[[d]]], {d,Most[Divisors[n]]}]; AppendTo[t,f], {n,2,100}]; t (* Second program: *) a[n_] := Product[Fibonacci[d]^MoebiusMu[n/d], {d, Divisors[n]}]; Array[a, 45] (* Jean-François Alcover, Jul 04 2019 *) -
PARI
a(n)=my(d=divisors(n)); fibonacci(n)/lcm(apply(fibonacci,d[1..#d-1])) \\ Charles R Greathouse IV, Oct 06 2016
Formula
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.
a(n) = Product_{d|n} Fib(d)^mu(n/d), where mu = A008683 (Bliss, Fulan, Lovett, Sommars, eq. (7)). - Jonathan Sondow, Jun 11 2013
a(n) = lcm(Fib(1),Fib(2),...,Fib(n))/lcm(Fib(1),Fib(2),...,Fib(n-1)). - Thomas Ordowski, Aug 03 2015
a(n) = Product_{k=1..n} Fib(gcd(n,k))^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} Fib(n/gcd(n,k))^(mu(gcd(n,k))/phi(n/gcd(n,k))) where mu = A008683, phi = A000010. - Richard L. Ollerton, Nov 08 2021
Extensions
More terms from Vladeta Jovovic, Nov 09 2001
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 29 2007
Edited by Charles R Greathouse IV, Oct 28 2009
Comments