A005479 -id:A005479 - cp's OEIS frontend

A245472 Number of composite Lucas numbers between the prime Lucas numbers A005479(n) and A005479(n+1).

0, 1, 0, 1, 0, 2, 1, 2, 0, 1, 11, 5, 3, 5, 5, 7, 9, 7, 33, 199, 39, 149, 109, 3, 245, 233, 263, 3425, 5, 1057, 1889, 725, 2223, 1559, 1711, 485, 5019, 15979, 1329, 7727, 6661, 4833, 25667, 8177, 4973, 45233, 8033, 11429, 23567, 10111, 9465, 141625, 43139, 56175

Offset: 1

Michel Lagneau, Jul 23 2014

nonn

			a(1)=0 because A005479(1)= 2, A005479(2)= 3 and there are no composite Lucas number between 2 and 3 (the number 1 is not composite).
a(6)=2 because A005479(6)= 47, A005479(7)= 199 and there are 2 composite Lucas numbers between 47 and 199: 76 = 2^2*19 and 123 = 3*41.

Jens Kruse Andersen, Table of n, a(n) for n = 1..63

Cf. A000032, A001606, A245470.

with(combinat,fibonacci):with(numtheory):i:=0:a:=n->2*fibonacci(n-1)+fibonacci(n):for n from 3 to 100 do:if type (a(n), prime)=true then printf(`%d, `, i):i:=0:else i:=i+1:fi:od:

a(n) = A001606(n+1) - A001606(n) - 1, for n > 1.

A001606 Indices of prime Lucas numbers.

Original entry on oeis.org

0, 2, 4, 5, 7, 8, 11, 13, 16, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113, 313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851, 7741, 8467, 10691, 12251, 13963, 14449, 19469, 35449, 36779, 44507, 51169, 56003, 81671, 89849, 94823, 140057, 148091, 159521, 183089, 193201, 202667, 344293, 387433, 443609, 532277, 574219, 616787, 631181, 637751, 651821, 692147, 901657, 1051849

Offset: 1

N. J. A. Sloane

Some of the larger entries may only correspond to probable primes.
Since (as noted under A000032) L(n) divides L(mn) whenever m is odd, L(n) cannot be prime unless n is itself prime, or else n contains no odd divisor, i.e., is a power of 2. Potential divisors of L(n) must satisfy certain linear forms dependent upon the parity of n, as shown in Vajda (1989), p. 82 (with a slight typographical error in the proof). - John Blythe Dobson, Oct 22 2007
Powers of 2 in this sequence are 2, 4, 8, 16; for 5 <= m <= 24, L(2^m) is composite; no factors of L(2^m) are known for m = 25, 26, 27, 29, 32, 33... (See Link section). - Serge Batalov, May 30 2017
2316773 is in the sequence, but its position is not yet defined. L(2316773) is a 484177-digit PRP. - Serge Batalov, Jun 11 2017

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 246.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. Vajda, Fibonacci and Lucas numbers and the Golden Section: Theory and Applications. Chichester: Ellis Horwood Ltd., 1989.

J. Brillhart, P. L. Montgomery and R. D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
David Broadhurst, Lucas record follows Fibonacci, Yahoo! group "primenumbers", Apr 26 2001.
David Broadhurst, Lucas record follows Fibonacci [Cached copy].
C. K. Caldwell, The Prime Glossary, Lucas prime.
H. Dubner and W. Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
Blair Kelly, Factorizations of Lucas numbers
Alex Kontorovich and Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
H. Lifchitz and R. Lifchitz, PRP Top Records, L(n).
Mersenneforum, A collection of factors of L(2^m).
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.
The Prime Database, V(81671), submitted Sep. 7, 2013, updated May 20, 2023.
Lawrence Somer and Michal Křížek, On Primes in Lucas Sequences, Fibonacci Quart. 53 (2015), no. 1, 2-23.
Eric Weisstein's World of Mathematics, Lucas Number.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.

Cf. A000032, A000204, A001605, A005479.
Cf. A080327 (n for which Lucas(n) and Fibonacci(n) are both prime).
Subsequence of A076697 (indices for which gpf(A000032(n)) sets a new record).

Reap[For[k = 0, k < 20000, k++, If[PrimeQ[LucasL[k]], Print[k]; Sow[k]]] ][[2, 1]] (* Jean-François Alcover, Feb 27 2016 *)

is(n)=ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Apr 24 2015

4 more terms from David Broadhurst, Jun 08 2001
More terms from T. D. Noe, Feb 15 2003 and Mar 04 2003; see link to The Prime Glossary.
387433, 443609, 532277 and 574219 found by Renaud Lifchitz, contributed by Eric W. Weisstein, Nov 29 2005
616787, 631181, 637751, 651821, 692147 found by Henri Lifchitz, circa Oct 01 2008, contributed by Alexander Adamchuk, Nov 28 2008
901657 and 1051849 found by Renaud Lifchitz, circa Nov 2008 and Mar 2009, contributed by Alexander Adamchuk, May 15 2010
1 more term from Serge Batalov, Jun 11 2017

A134809 Cyclops primes.

Original entry on oeis.org

101, 103, 107, 109, 307, 401, 409, 503, 509, 601, 607, 701, 709, 809, 907, 11027, 11047, 11057, 11059, 11069, 11071, 11083, 11087, 11093, 12011, 12037, 12041, 12043, 12049, 12071, 12073, 12097, 13033, 13037, 13043, 13049, 13063

Offset: 1

Omar E. Pol, Nov 25 2007

Cyclops numbers that are prime numbers: primes with an odd number of digits with middle digit 0 that have only one digit 0.
The only known Fibonacci number in this sequence is 99194853094755497 (see A005478 and A182809).
The only known Lucas number in this sequence is 688846502588399 (see A005479 and A182811).

Harvey P. Dale, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 688846502588399
G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 99194853094755497

Intersection of prime numbers A000040 and cyclops numbers A134808.

Cf. A005478, A005479, A056709, A069675, A182809, A182811.

(* First run the program given for A134808 *) Select[Prime[Range[2000]], cyclopsQ] (* Alonso del Arte, Dec 16 2010 *)
cycQ[n_]:=Module[{idn=IntegerDigits[n],len},len=Length[idn];OddQ[len] && Count[idn,0] == 1 && idn[[(len+1)/2]]==0]; Select[Flatten[Table[Prime[ Range[ PrimePi[10^(2n)+1],PrimePi[10^(2n+1)]]],{n,2}]],cycQ] (* Harvey P. Dale, Jun 20 2014 *)

# cyclops() in A134808
from sympy import isprime
print([c for c in cyclops(upto=13063) if isprime(c)]) # Michael S. Branicky, Jan 05 2021

Links added by Omar E. Pol, Mar 25 2011

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

A153867 Primes which are the sum of four consecutive Fibonacci numbers.

Original entry on oeis.org

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

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

1+1+2+3=7, 1+2+3+5=11, 3+5+8+13=29, 5+8+13+21=47, 21+34+55+89=199,...

Vincenzo Librandi, Table of n, a(n) for n = 1..28
Harvey P. Dale and others, A005479 and A153867, SeqFan list, Apr 24 2014.

Cf. A001906, A000071, A001605, A153862, A153863, A153865, A153866.

Primes >3 of A000032 and of A000204.

a=0;b=1;c=1;lst={};Do[d=Fibonacci[n];p=a+b+c+d;If[PrimeQ[p],AppendTo[lst,p]];a=b;b=c;c=d,{n,3,6!}];lst
Select[Total/@Partition[Fibonacci[Range[500]],4,1],PrimeQ] (* Harvey P. Dale, Apr 23 2014 *)

list(lim)=my(v=List(),a=7,b=11); while(a<=lim, if(isprime(a), listput(v,a)); [a,b]=[b,a+b]); Vec(v) \\ Charles R Greathouse IV, Oct 07 2016

a(n) = A005479(n+2). - R. J. Mathar, Jan 05 2009 [See also link to the SeqFan list. - M. F. Hasler, Apr 24 2014]

One additional term (a(19)) from Harvey P. Dale, Apr 23 2014

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

A121533 Fibonacci-Lucas prime twins: Prime Fibonacci numbers Fibonacci(k) such that Lucas numbers Lucas(k) = Fibonacci(k-1) + Fibonacci(k+1) are also prime.

Original entry on oeis.org

3, 5, 13, 89, 233, 1597, 2971215073

Offset: 1

Alexander Adamchuk, Aug 05 2006

nonn

Indices for Fibonacci-Lucas prime twins are A080327(n) = {4, 5, 7, 11, 13, 17, 47, ...}. Corresponding Lucas-Fibonacci prime twins are A121534(n) = {7, 11, 29, 199, 521, 3571, 6643838879, ...}. 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) = 3 because Fibonacci(4) = 3 is prime and Lucas(4) = 5 is also prime.

Cf. A121534, 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}]

cp's OEIS Frontend

A245472 Number of composite Lucas numbers between the prime Lucas numbers A005479(n) and A005479(n+1).

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A001606 Indices of prime Lucas numbers.

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A134809 Cyclops primes.

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A285992 Primes in the bisected Lucas sequence A002878.

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A153867 Primes which are the sum of four consecutive Fibonacci numbers.

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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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A052012 Number of primes between successive Lucas numbers.

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