A061447 -id:A061447 - cp's OEIS frontend

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

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

A061446 Primitive part of Fibonacci(n).

Original entry on oeis.org

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

A061442 Numbers k such that the Lucas Aurifeuillian primitive part A of Lucas(k) is prime.

Original entry on oeis.org

25, 35, 45, 55, 65, 75, 85, 95, 105, 125, 145, 165, 185, 275, 335, 355, 535, 655, 735, 805, 925, 955, 1095, 1195, 1215, 1275, 1305, 1325, 1435, 1575, 1655, 1765, 2015, 2205, 2715, 2745, 2885, 3905, 3935, 4275, 5705, 5995, 7755, 8565, 11695, 11785, 11825, 11925

Offset: 1

David Broadhurst, Jun 10 2001

nonn

The Aurifeullian primitive part A of Lucas(k) only exists when k is a multiple of 5, hence all terms of this sequence are multiples of 5. - Sean A. Irvine, Feb 15 2023

David Broadhurst, Table of n, a(n) for n=1..83 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, A061443.

Edited by Arkadiusz Wesolowski, Dec 29 2012

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

A126015 Lucas numbers divided by their primitive part.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 4, 3, 1, 7, 1, 3, 44, 1, 1, 18, 1, 7, 116, 3, 1, 47, 11, 3, 76, 7, 1, 738, 1, 1, 796, 3, 319, 322, 1, 3, 2084, 47, 1, 5058, 1, 7, 25916, 3, 1, 2207, 29, 123, 14284, 7, 1, 5778, 2189, 47, 37396, 3, 1, 695842, 1, 3, 465044, 1, 5731, 237618, 1, 7, 256316

Offset: 1

T. D. Noe, Dec 16 2006

nonn

Note that a(n)=1 for prime n and for n=2^k for k=0,1,2,...

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A000032 (Lucas numbers), A105602 (Fibonacci numbers divided by their primitive part).

a(n)=L(n)/A061447(n)

A190757 Lucas Aurifeuillian primitive part A of Lucas(10*n - 5).

Original entry on oeis.org

1, 1, 101, 71, 181, 39161, 24571, 12301, 1158551, 87382901, 21211, 373270451, 28143378001, 32414581, 1322154751061, 9062194370461, 1550853481, 2819407321151, 265272771839851, 2366632711, 137083914639998701, 85417012034751151, 3455782010101

Offset: 1

Arkadiusz Wesolowski, Dec 29 2012

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..250
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), pp. 251-260, S1.
C. K. Caldwell, "Top Twenty" page, Lucas Aurifeuillian primitive part

Cf. A061447, A190781.

lst = {1}; n = 23; Do[f = LucasL[k]; Do[f = f/GCD[f, lst[[d]]], {d, Most@Divisors[k]}]; AppendTo[lst, f], {k, 2, 10*n - 5}]; Table[GCD[lst[[5*k]], 5*Fibonacci[k]*(Fibonacci[k] - 1) + 1], {k, 1, 2*n - 1, 2}]

a(n) = GCD(A061447(10*n-5), A027941(n-1)*A106729(n-1) + 1).

A190781 Lucas Aurifeuillian primitive part B of Lucas(10*n - 5).

Original entry on oeis.org

11, 31, 151, 911, 541, 3641, 272611, 18451, 12760031, 7947701, 767131, 4106261531, 28144128251, 34379101, 120196353941, 823837075741, 51164521, 4215154433351, 2918000731816531, 73998076231, 12462174208709101, 939587136717207031, 3467131047901

Offset: 1

Arkadiusz Wesolowski, Dec 29 2012

nonn

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..250
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), pp. 251-260, S1.
C. K. Caldwell, "Top Twenty" page, Lucas Aurifeuillian primitive part

Cf. A061447, A190757.

lst = {1}; n = 23; Do[f = LucasL[k]; Do[f = f/GCD[f, lst[[d]]], {d, Most@Divisors[k]}]; AppendTo[lst, f], {k, 2, 10*n - 5}]; Table[GCD[lst[[5*k]], 5*Fibonacci[k]*(Fibonacci[k] + 1) + 1], {k, 1, 2*n - 1, 2}]

a(n) = GCD(A061447(10*n-5), A032908(n)*A106729(n-1) + 1).

A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

Original entry on oeis.org

1, 2, 2, 7, 2, 1, 11, 3, 11, 1, 29, 47, 29, 1, 19, 41, 19, 1, 199, 23, 199, 1, 521, 281, 521, 1, 31, 2207, 31, 1, 3571, 107, 3571, 1, 9349, 2161, 9349, 1, 211, 13201, 211, 1, 64079, 1103, 64079, 1, 15251, 90481, 15251, 1, 5779, 14503, 5779, 1, 1149851, 2521

Offset: 1

Alexander Adamchuk, Aug 16 2006, Sep 21 2006

a(1) = 1 and a(4k+2) = 1 for k>0.

For k >1 a(4k-1) = a(4k+1) = A072183(2k+1) = A061447(2k+1) Primitive part of Lucas(n).

Cf. A000045, A000032, A079451, A001578, A096362, A058036, A121709, A090585.

Cf. A061447, A072183.

Table[Numerator[Sum[Fibonacci[k],{k,1,n}]/Product[Fibonacci[k],{k,1,n}]],{n,1,100}]
With[{fibs=Fibonacci[Range[60]]},Numerator[Accumulate[fibs]/Rest[ FoldList[ Times,1,fibs]]]] (* This is significantly faster than the first program above *) (* Harvey P. Dale, Aug 19 2012 *)

a(n) = numerator( sum(k=1..n, Fibonacci(k)) / prod(k=1..n, Fibonacci(k)) ).

cp's OEIS Frontend

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

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A061446 Primitive part of Fibonacci(n).

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A061442 Numbers k such that the Lucas Aurifeuillian primitive part A of Lucas(k) is prime.

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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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A126015 Lucas numbers divided by their primitive part.

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A190757 Lucas Aurifeuillian primitive part A of Lucas(10*n - 5).

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A190781 Lucas Aurifeuillian primitive part B of Lucas(10*n - 5).

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A121708 Numerator of Sum/Product of first n Fibonacci numbers A000045[n].

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