A061442 -id:A061442 - cp's OEIS frontend

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

David Broadhurst, Jun 10 2001

nonn

Fib(n) = A000045(n) = Product_{d|n} a(d), Lucas(n) = A000204(n) = Product_{d|2n and 2^m|d iff 2^m|2n} 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 GCD-morphic sequences F, i.e., F 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 a certain natural number-valued sequence. - Maciej Dziemianczuk, Jan 15 2009
This is the LCM-transform of the Fibonacci numbers (cf. Nowicki). - N. J. A. Sloane, Jan 02 2016

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.

Cf. A008683, A061447, A061254, A061445, A061442, A061443, A105602, A126025, A126069.

Cf. A000010 (comments on product formulas).

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

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 *)

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

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

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

A061447 Primitive part of Lucas(n).

Original entry on oeis.org

1, 3, 4, 7, 11, 6, 29, 47, 19, 41, 199, 46, 521, 281, 31, 2207, 3571, 321, 9349, 2161, 211, 13201, 64079, 2206, 15251, 90481, 5779, 101521, 1149851, 2521, 3010349, 4870847, 9901, 4250681, 64681, 103681, 54018521, 29134601, 67861, 4868641, 370248451

Offset: 1

David Broadhurst, Jun 10 2001

nonn

T. D. Noe, Table of n, a(n) for n=1..1000
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260, S1-S15. Math. Rev. 89h:11002.
C. K. Caldwell, Lucas Aurifeuillian primitive part

Cf. A061446, A061254, A061445, A061442, A061443.

Cf. A126015

t={1}; Do[f=LucasL[n]; Do[f=f/GCD[f,t[[d]]], {d,Most[Divisors[n]]}]; AppendTo[t,f], {n,2,100}]; t

Primitive part of L(n) is primitive part of F(2n).

a(n) = Product_{ d divides 2*n } Fibonacci(2*n/d)^mu(d). - Vladeta Jovovic, Mar 08 2004

More terms from Vladeta Jovovic, Mar 08 2004

A061254 Odd composite numbers n such that primitive part of Fibonacci(n) (see A061446) is prime.

Original entry on oeis.org

9, 15, 21, 33, 35, 39, 45, 51, 63, 65, 75, 93, 105, 111, 119, 121, 123, 135, 145, 185, 195, 201, 207, 209, 225, 231, 235, 245, 285, 287, 299, 301, 321, 335, 363, 399, 423, 453, 473, 693, 707, 771, 1047, 1113, 1215, 1365, 1371, 1387, 1533, 1537

Offset: 1

David Broadhurst, Jun 10 2001

nonn

David Broadhurst, Table of n, a(n) for n=1..151 yielding primes or probable primes
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260, S1-S15. Math. Rev. 89h:11002.

Cf. A061446, A061447, A061445, A061442, A061443.

A061446[n_] := Product[Fibonacci[d]^MoebiusMu[n/d], {d, Divisors[n]}];
Select[Range[1, 1999, 2], CompositeQ[#] && PrimeQ[A061446[#]]&] (* Jean-François Alcover, Jul 04 2019 *)

Definition corrected by T. D. Noe, Dec 14 2006

A061443 Numbers k such that the Lucas Aurifeuillian primitive part B of Lucas(k) is prime.

Original entry on oeis.org

5, 15, 25, 35, 45, 75, 85, 105, 145, 155, 165, 185, 225, 255, 305, 315, 325, 335, 355, 365, 375, 475, 485, 525, 565, 575, 635, 695, 715, 765, 885, 1235, 1325, 1375, 1515, 2255, 2285, 3085, 3185, 3355, 3565, 3745, 3885, 4325, 4995, 5525, 5915, 6195, 6565, 6975, 6995, 7785, 8855

Offset: 1

David Broadhurst, Jun 10 2001

nonn

David Broadhurst, Table of n, a(n) for n=1..93 yielding primes or probable primes
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260, S1-S15. Math. Rev. 89h:11002.
C. K. Caldwell, Lucas Aurifeuillian primitive part

Cf. A061446, A061447, A061254, A061445, A061442.

Corrected by T. D. Noe, Dec 17 2006

Edited by Arkadiusz Wesolowski, Dec 29 2012

A061445 Composite numbers n such that primitive part of Lucas(n) (see A061447) is prime.

Original entry on oeis.org

9, 10, 14, 15, 16, 20, 21, 26, 27, 30, 33, 36, 38, 49, 56, 62, 66, 68, 70, 72, 76, 78, 80, 86, 90, 91, 110, 117, 120, 121, 136, 140, 144, 164, 168, 172, 178, 202, 207, 220, 261, 284, 328, 354, 357, 420, 423, 458, 459, 468, 480, 504, 513, 530, 586, 606

Offset: 1

David Broadhurst, Jun 10 2001

nonn

			16 is a term because A061447(16)=2207 is a prime. - _Sean A. Irvine_, Feb 15 2023

David Broadhurst, Table of n, a(n) for n = 1..258 yielding primes or probable primes
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260, S1-S15. Math. Rev. 89h:11002.
C. K. Caldwell, Lucas Aurifeuillian primitive part

Cf. A061446, A061447, A061254, A061442, A061443.

Definition corrected by T. D. Noe, Dec 14 2006

Missing a(5)=16 inserted by Sean A. Irvine, Feb 15 2023

A153443 Aurifeuillian primes of the form 2^k+1.

Original entry on oeis.org

3, 5, 11, 13, 17, 43, 241, 257, 331, 683, 2731, 5419, 43691, 61681, 65537, 174763, 2796203, 15790321, 18837001, 22366891, 715827883, 4278255361, 4562284561, 77158673929, 1133836730401, 2932031007403, 4363953127297

Offset: 1

Artur Jasinski, Dec 26 2008

nonn

Take an irreducible real factor of x^k+1 and substitute x=2. If the result is a prime then it belongs in this sequence. For example for k=5 the polynomial x^5+1=(x+1)(x^4-x^3+x^2-x+1) and substituting x->2 in (x^4-x^3+x^2-x+1) we get the prime number 11. So 11 is a term. [Clarified by N. J. A. Sloane, Jul 03 2020]

A061442.

A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

Original entry on oeis.org

10, 14, 18, 22, 26, 30, 42, 54, 58, 66, 70, 86, 94, 98, 106, 110, 126, 130, 138, 146, 158, 174, 186, 210, 222, 226, 258, 302, 334, 434, 462, 478, 482, 522, 566, 602, 638, 706, 734, 750, 770, 782, 914, 1062, 1086, 1114, 1126, 1226, 1266, 1358, 1382, 1434, 1742, 1926

Offset: 1

Eric Chen, Jan 18 2015

nonn

All terms are congruent to 2 modulo 4.
Phi_n(x) is the n-th cyclotomic polynomial.
Numbers n such that Phi_{2nL(n)}(2) is prime.
Let J(n) = 2^n+1, J*(n) = the primitive part of 2^n+1, this is Phi_{2n}(2).
Let L(n) = the Aurifeuillian L-part of 2^n+1, L(n) = 2^(n/2) - 2^((n+2)/4) + 1 for n congruent to 2 (mod 4).
Let L*(n) = GCD(L(n), J*(n)).
This sequence lists all n such that L*(n) is prime.

			14 is in this sequence because the left Aurifeuillian primitive part of 2^14+1 is 113, which is prime.
34 is not in this sequence because the left Aurifeuillian primitive part of 2^34+1 is 130561, which equals 137 * 953 and is not prime.

Eric Chen, Gord Palameta, Factorization of Phi_n(2) for n up to 1280
Samuel Wagstaff, The Cunningham project
Eric W. Weisstein's World of Mathematics, Aurifeuillean Factorization.

Cf. A250198, A153443, A019320, A072226, A161508, A092440, A229767, A061442.

Select[Range[2000], Mod[#, 4] == 2 && PrimeQ[GCD[2^(#/2) - 2^((#+2)/4) + 1, Cyclotomic[2*#, 2]]] &]

isok(n) = isprime(gcd(2^(n/2) - 2^((n+2)/4) + 1, polcyclo(2*n, 2))); \\ Michel Marcus, Jan 27 2015

cp's OEIS Frontend

A061446 Primitive part of Fibonacci(n).

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A061447 Primitive part of Lucas(n).

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A061254 Odd composite numbers n such that primitive part of Fibonacci(n) (see A061446) is prime.

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A061443 Numbers k such that the Lucas Aurifeuillian primitive part B of Lucas(k) is prime.

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A061445 Composite numbers n such that primitive part of Lucas(n) (see A061447) is prime.

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A153443 Aurifeuillian primes of the form 2^k+1.

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A250197 Numbers k such that the left Aurifeuillian primitive part of 2^k+1 is prime.

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Mathematica

PARI