

A061446


Primitive part of Fibonacci(n).


28



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;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

Fib(n) = A000045(n) = Product_{dn} a(d), Lucas(n) = A000204(n) = Product_{d2n and 2^md iff 2^m2n} a(d) (e.g., Lucas(4) = 7 = a(8), Lucas(6) = 18 = a(12)*a(4)).  Len Smiley, 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 the characterization of the family of all GCDmorphic sequences F, i.e., F such that GCD(F(m),F(n)) = F(GCD(m,n)), was posed by A. K. Kwasniewski (GCDmorphic Problem). Dziemianczuk and Bajguz (2008) showed that any GCDmorphic sequence is coded by a certain natural numbervalued sequence.  Maciej Dziemianczuk, Jan 15 2009
This is the LCMtransform of the Fibonacci numbers (cf. Nowicki).  N. J. A. Sloane, Jan 02 2016


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), 519536.
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), pp. 251260, S1S15. 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) 3048.
M. Dziemianczuk and W. Bajguz, On GCDmorphic sequences, arXiv:0802.1303 [math.CO], 2008.
A. K. Kwasniewski, Cobweb posets as noncommutative prefabs, Adv. Stud. Contemp. Math. vol.14 (1) 2007. pp. 3747.
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 LCMsequences, arXiv:1310.2416 [math.NT], 2013.
A. Nowicki, Strong divisibility and LCMsequences, Am. Math. Mnthly 122 (2015), 958966.


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 nth cyclotomic polynomial and phi is Euler's totient function A000010.
a(n) = Product_{dn} 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(n1)).  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


MAPLE

N:= 200; # to get a(1) to a(N)
L[0]:= 1:
for i from 1 to N do L[i]:=ilcm(L[i1], combinat:fibonacci(i)) od:
seq(L[i]/L[i1], i=1..N); # Robert Israel, Aug 03 2015


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] (* JeanFrançois Alcover, Jul 04 2019 *)


PROG

(PARI) a(n)=my(d=divisors(n)); fibonacci(n)/lcm(apply(fibonacci, d[1..#d1])) \\ Charles R Greathouse IV, Oct 06 2016


CROSSREFS

Cf. A008683, A061447, A061254, A061445, A061442, A061443, A105602, A126025, A126069.
Cf. A000010 (comments on product formulas).
Sequence in context: A271862 A309373 A131401 * A280690 A240000 A193770
Adjacent sequences: A061443 A061444 A061445 * A061447 A061448 A061449


KEYWORD

nonn


AUTHOR

David Broadhurst, Jun 10 2001


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


STATUS

approved



