A001606 -id:A001606 - cp's OEIS frontend

A014554 Erroneous version of A001606.

2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353

Offset: 0

dead

A005479 Prime Lucas numbers (cf. A000032).

2, 3, 7, 11, 29, 47, 199, 521, 2207, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521

Offset: 1

N. J. A. Sloane

It appears that a(n) is the intersection ( or a subset of the intersection ) of A113192[n], Primes that are the difference of two Lucas numbers and A113188[n], Primes that are the difference of two Fibonacci numbers, excluding A113192[1] = A113188[1] = 2. - Alexander Adamchuk, Aug 06 2006
For n>2 also: Primes which are the sum of four consecutive Fibonacci numbers, a(n) = A153867(n-2), cf. link to SeqFan list (Apr. 2014). - M. F. Hasler, Apr 24 2014
Conjectures: 7, 47 and 2207 are the only a(n) mod 10 = 7. They are also the only a(n) values where the Lucas index is not a prime.  See A001606 for the Lucas index values of these primes. See A266587 for the divisibility of Lucas numbers by powers of primes.  - Richard R. Forberg, Mar 24 2016

R. K. Guy, Unsolved Problems in Number Theory, Section A3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..28
J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
Harvey P. Dale and others, A005479 and A153867, SeqFan list, Apr 24 2014.
Blair Kelly, Factorizations of Lucas numbers
Ron Knott, The First 200 Lucas numbers and their factors.
Tony D. Noe and Jonathan Vos Post, Primes in Fibonacci n-step and Lucas n-step Sequences, J. of Integer Sequences, Vol. 8 (2005), Article 05.4.4.
Eric Weisstein's World of Mathematics, Lucas Number

Cf. A000032, A001606, A113192, A113188.

Select[LucasL[Range[0,250]], PrimeQ] (* Harvey P. Dale, Nov 02 2011 *)

One further term (from the Knott web site) from Parthasarathy Nambi, Jun 27 2008

A285992 Primes in the bisected Lucas sequence A002878.

Original entry on oeis.org

11, 29, 199, 521, 3571, 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149, 412670427844921037470771, 258899611203303418721656157249445530046830073044201152332257717521

Offset: 1

R. J. Mathar, Apr 30 2017

nonn

Subsequence of A005479.

Cf. A001606, A002878, A005479, A121534, A180363, A269254, A294099 (row 3).

select(isprime, [seq(combinat:-fibonacci(2*n)+combinat:-fibonacci(2*n+2), n=1..200)]); # Robert Israel, May 01 2017

Select[LucasL[Range[1, 400, 2]], PrimeQ] (* Vincenzo Librandi, May 01 2017 *)
Select[LinearRecurrence[{3,-1},{1,4},160],PrimeQ] (* Harvey P. Dale, Sep 01 2024 *)

A002878 INTERSECT A000040.

A000032(k) for odd k in A001606. - Robert Israel, May 01 2017

A117522 Numbers k such that L(2*k + 1) is prime, where L(m) is a Lucas number.

Original entry on oeis.org

2, 3, 5, 6, 8, 9, 15, 18, 20, 23, 26, 30, 35, 39, 56, 156, 176, 251, 306, 308, 431, 548, 680, 2393, 2396, 2925, 3870, 4233, 5345, 6125, 6981, 7224, 9734, 17724, 18389, 22253, 25584, 28001, 40835, 44924, 47411, 70028, 74045, 79760, 91544, 96600, 101333, 172146, 193716, 221804, 266138, 287109, 308393, 315590, 318875, 325910, 346073, 450828, 525924

Offset: 1

Parthasarathy Nambi, Apr 26 2006

For n = 24..43, we can only claim that L(2*a(n) + 1) is a probable prime. Sequence arises in a study of A269254; for detailed theory, see [Hone]. - L. Edson Jeffery, Feb 09 2018

			If k = 56, then L(2*k + 1) is a prime with twenty-four digits.

Andrew N. W. Hone, et al., On a family of sequences related to Chebyshev polynomials, arXiv:1802.01793 [math.NT], 2018.

Cf. A000032, A001606, A269251, A269252, A269253, A269254.

Cf. A294099, A298675, A298677, A298878, A299045, A299071, A285992, A299107, A299109, A088165, A299100, A299101, A113501.

Values beyond 680 from L. Edson Jeffery, et al., Feb 02 2018
a(44)-a(56) from Robert Price, Jun 12 2025
a(57)-a(59) (using data in A001606) from Alois P. Heinz, Jun 12 2025

A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 19, 29, 31, 47, 53, 89, 131, 139, 199, 233, 521, 607, 953, 1453, 1597, 2207, 2351, 2579, 3571, 6763, 9349, 10891, 28513, 28649, 28657, 42187, 44771, 46279, 75017, 189653, 317777, 514229, 1981891, 2177699, 3010349, 3206767

Offset: 1

T. D. Noe, Oct 17 2005

nonn

The difference F(i)-F(j) equals the sum F(j-1)+...+F(i-2) [Corrected by Patrick Capelle, Mar 01 2008]. In general, we need gcd(i,j)=1 for F(i)-F(j) to be prime. The exceptions are handled by the following rule: if i and j are both even or both odd, then F(i)-F(j) is prime if either (1) i-j=4 and L(i-2) is a Lucas prime or (2) i-j=2 and F(i-1) is a Fibonacci prime.

			The prime 139 is here because it is F(12)-F(5).

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

Cf. A000045 (Fibonacci numbers), A001605 (Fibonacci(n) is prime), A001606 (Lucas(n) is prime), A113189 (number of times that Fibonacci(n)-Fibonacci(i) is prime for i=0..n-3).

lst={}; Do[p=Fibonacci[n]-Fibonacci[i]; If[PrimeQ[p], AppendTo[lst, p]], {n, 2, 40}, {i, n-1}]; Union[lst]
Select[Union[Flatten[Differences/@Subsets[Fibonacci[Range[50]],{2}]]],PrimeQ] (* Harvey P. Dale, Aug 04 2024 *)

list(lim)=my(v=List(),F=vector(A130233(lim),i,fibonacci(i)),s,t); for(i=1,#F, s=0; forstep(j=i,1,-1, s+=F[j]; if(s>lim, break); if(isprime(s), listput(v,s)))); Set(v) \\ Charles R Greathouse IV, Oct 07 2016

A079451 Largest prime dividing the n-th Lucas number (A000032); 1 when no such prime exists.

Original entry on oeis.org

2, 1, 3, 2, 7, 11, 3, 29, 47, 19, 41, 199, 23, 521, 281, 31, 2207, 3571, 107, 9349, 2161, 211, 307, 461, 1103, 151, 90481, 5779, 14503, 19489, 2521, 3010349, 4481, 9901, 63443, 911, 103681, 54018521, 29134601, 859, 3041, 370248451, 1427, 144481, 967, 541, 275449

Offset: 0

Lekraj Beedassy, Jan 13 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000 (using Blair Kelly's data)
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.
Blair Kelly, Factorizations of first 1000 Lucas numbers.

Cf. A000032 (Lucas numbers), A006530 (greatest prime factor).
Cf. A001606 (indices of prime Lucas numbers <=> where a(n) = A000032(n)), subsequence of A076697 (indices of record values in this sequence).
Cf. A280104 (same for smallest prime factor).

[2,1] cat [Maximum(PrimeDivisors(Lucas(n))): n in [2..60]]; // Vincenzo Librandi, Dec 26 2016

A079451 := proc(n)
        A006530(A000032(n)) ;
end proc:
seq(A079451(n),n=0..30) ; # R. J. Mathar, Oct 26 2013
# second Maple program:
a:= n-> max(ifactors((<<1|1>, <1|0>>^n. <<2, -1>>)[1, 1])[2][..., 1][], 1):
seq(a(n), n=0..46);  # Alois P. Heinz, Aug 04 2025

Join[{2,1},f[n_]:=(FactorInteger@LucasL@n)[[-1,1]];Array[f,60,2]] (* Vincenzo Librandi, Dec 26 2016 *)

a(n) = my(f = factor(fibonacci(n+1)+fibonacci(n-1))); if (om = #f~, f[om, 1], 1); \\ Michel Marcus, Oct 26 2013

A079451(n) = A006530(A000032(n)) \\ M. F. Hasler, Apr 10 2025

def A079451(n): return A006530(A000032(n)) # M. F. Hasler, Apr 10 2025

a(n) = A006530(A000032(n)). - Felix Fröhlich, Dec 26 2016

More terms from Michel Marcus, Oct 26 2013

Modified b-file and scripts so that a(1)=1. - David Radcliffe, Aug 03 2025

A080327 Numbers k for which Lucas(k) and Fibonacci(k) are both prime.

Original entry on oeis.org

4, 5, 7, 11, 13, 17, 47, 148091

Offset: 1

T. D. Noe, Feb 15 2003

The intersection of A001605 and A001606. Fibonacci(148091) and Lucas(148091) are probable primes.
Corresponding Fibonacci-Lucas prime twins are listed in A121533. Corresponding Lucas-Fibonacci prime twins are listed in A121534. Fibonacci(148091) and Lucas(148091) are probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits. - Alexander Adamchuk, Aug 05 2006
Heuristically, this sequence is finite. It is quite probable, but presently unprovable, that it is now complete. - David Broadhurst, Jun 25 2008
Western Number Theory problem 007:13 by Gary Walsh asks to prove that a(8) = 148091 is in this sequence. - Charles R Greathouse IV, May 21 2014

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 246.

David Broadhurst, Prime Curio: 10660...49391 (61899-digits)
Gerry Myerson, Western Number Theory Problems, 17 & 19 Dec 2007.

Cf. A001605, A001606, A121533, A121534, A000045, A005478, A000032, A000204, A005479, A080327.

Select[Range[0, 100], PrimeQ[Fibonacci[#]] && PrimeQ[LucasL[#]] & ] (* Robert Price, May 27 2019 *)

is(n)=isprime(n) && ispseudoprime(fibonacci(n)) && ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, May 21 2014

A121534 Lucas-Fibonacci prime twins: Prime Lucas numbers Lucas(k) such that Fibonacci numbers Fibonacci(k) are also prime.

Original entry on oeis.org

7, 11, 29, 199, 521, 3571, 6643838879

Offset: 1

Alexander Adamchuk, Aug 05 2006

nonn

Indices for Lucas-Fibonacci prime twins are A080327(n). Corresponding Fibonacci-Lucas prime twins are A121533(n). Probable primes Fibonacci(148091) and Lucas(148091) are the next probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits.

			a(1) = 7 because Lucas(4) = 7 is prime and Fibonacci(4) = 3 is prime too.

Cf. A121533, A000045, A005478, A001605, A001606, A000032, A000204, A005479, A080327.

Do[f=Fibonacci[n]; l=Fibonacci[n-1]+Fibonacci[n+1]; If[PrimeQ[f]&&PrimeQ[l], Print[{f,l}]], {n,10000}]
nn=1000;Transpose[Select[Thread[{Fibonacci[Range[nn]], LucasL[ Range[nn]]}],And@@PrimeQ[#]&]][[2]] (* Harvey P. Dale, Jul 08 2011 *)

a(1) and example corrected by Harvey P. Dale, Jul 08 2011

A052012 Number of primes between successive Lucas numbers.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 4, 6, 9, 15, 20, 31, 48, 72, 110, 170, 257, 400, 608, 950, 1448, 2256, 3487, 5413, 8440, 13118, 20478, 31932, 49995, 78222, 122553, 192262, 301826, 474039, 745772, 1173270, 1848000, 2912623, 4593723, 7249438, 11448047

Offset: 1

Patrick De Geest, Nov 15 1999

			Between L(7)=29 and L(8)=47 we find the following primes: 31, 37, 41 and 43 hence a(7)=4.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000032, A000204, A000720, A001606, A005479, A052011, A277062.

a052012 n = a052012_list !! (n-1)
a052012_list = c 1 0 $ tail a000204_list where
  c x y ls'@(l:ls) | x < l     = c (x+1) (y + a010051 x) ls'
                   | otherwise = y : c (x+1) 0 ls
-- Reinhard Zumkeller, Dec 18 2011

PrimePi[Last[#]-1]-PrimePi[First[#]]&/@Partition[LucasL[ Range[45]],2,1] (* Harvey P. Dale, Jun 28 2011 *)

a(n) = pi(L(n + 1) - 1) - pi(L(n)), where pi is the prime counting function (A000720) and L = A000032. - Wesley Ivan Hurt, Nov 09 2023

a(n) = A277062(n+1) - A277062(n) - [n+1 in A001606], where [] denotes the Iverson bracket. - Amiram Eldar, Jun 10 2024

A277062 Number of primes <= n-th Lucas number.

Original entry on oeis.org

1, 0, 2, 2, 4, 5, 7, 10, 15, 21, 30, 46, 66, 98, 146, 218, 329, 500, 757, 1158, 1766, 2716, 4164, 6420, 9907, 15320, 23760, 36878, 57356, 89288, 139283, 217506, 340059, 532321, 834147, 1308186, 2053958, 3227229, 5075229, 7987852, 12581575, 19831014

Offset: 0

Vincenzo Librandi, Nov 09 2016

nonn

Amiram Eldar, Table of n, a(n) for n = 0..101 (calculated using Kim Walisch's primecount)
Kim Walisch, Fast C++ prime counting function implementation (primecount).

Cf. A000032, A000720, A001606, A005479, A052012, A054782.

Magma
```
[#PrimesUpTo(Lucas(n)): n in [0..41]];
```

a:= n-> numtheory[pi]((<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]):
seq(a(n), n=0..35);  # Alois P. Heinz, Nov 09 2016

Mathematica
```
Table[PrimePi[LucasL[n]], {n, 0, 50}]
```

a(n) = A000720(A000032(n)). - Michel Marcus, Jun 10 2024

cp's OEIS Frontend

A014554 Erroneous version of A001606.

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A005479 Prime Lucas numbers (cf. A000032).

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A285992 Primes in the bisected Lucas sequence A002878.

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A117522 Numbers k such that L(2*k + 1) is prime, where L(m) is a Lucas number.

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A113188 Primes that are the difference of two Fibonacci numbers; primes in A007298.

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A079451 Largest prime dividing the n-th Lucas number (A000032); 1 when no such prime exists.

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A080327 Numbers k for which Lucas(k) and Fibonacci(k) are both prime.

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A121534 Lucas-Fibonacci prime twins: Prime Lucas numbers Lucas(k) such that Fibonacci numbers Fibonacci(k) are also prime.

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