A080327 -id:A080327 - cp's OEIS frontend

A001606 Indices of prime Lucas numbers.

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

A072381 Numbers m such that Fibonacci(m) is a semiprime.

Original entry on oeis.org

8, 9, 10, 14, 19, 22, 26, 31, 34, 41, 53, 59, 61, 71, 73, 79, 89, 94, 101, 107, 109, 113, 121, 127, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 397, 401, 467, 587, 599, 601, 613, 631, 653, 743, 991, 1091, 1223, 1373, 1487

Offset: 1

Shyam Sunder Gupta, Jul 20 2002

Note that there are two cases: (1) n is 2p, in which case the semiprime is Fibonacci(p)*Lucas(p) for some prime p, or (2) n is a power of a prime p^k for k > 0. In the first case, the primes p are in sequence A080327. In the second case, it appears that k=1 except for n = 8, 9 and 121. - T. D. Noe, Sep 23 2005
The associated sequence of Fibonacci numbers contains no squares, since the only Fibonacci numbers which are square are 1 and 144. Consequently this is a subsequence of A114842. - Charles R Greathouse IV, Sep 24 2012
Sequence continues as 1543?, 1709, 1741?, 1759, 1801?, 1889, 1987, ..., where ? marks uncertain terms. - Max Alekseyev, Jul 10 2016

			a(4) = 14 because the 14th Fibonacci number 377 = 13*29 is a semiprime.

Y. Bugeaud, F. Luca, M. Mignotte and S. Siksek, On Fibonacci numbers with few prime divisors, Proc. Japan Acad., 81, Ser. A (2005), pp. 17-20.
Ron Knott, Fibonacci numbers
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A053409, A085726 (n such that n-th Lucas number is a semiprime).

Column k=2 of A303215.

Select[Range[200], Plus@@Last/@FactorInteger[Fibonacci[ # ]] == 2&] (Noe)
Select[Range[1500],PrimeOmega[Fibonacci[#]]==2&] (* Harvey P. Dale, Dec 13 2020 *)

for(n=2,9999,bigomega(fibonacci(n))==2&&print1(n",")) \\ - M. F. Hasler, Oct 31 2012

issemi(n)=bigomega(n)==2
is(n)=if(n%2, my(p); if(issquare(n,&p), isprime(p) && isprime(fibonacci(p)) && isprime(fibonacci(n)/fibonacci(p)), isprime(n) && issemi(fibonacci(n))), (isprime(n/2) && isprime(fibonacci(n/2)) && isprime(fibonacci(n)/fibonacci(n/2))) || n==8) \\ Charles R Greathouse IV, Oct 06 2016

More terms from Don Reble, Jul 31 2002
a(49)-a(50) from Max Alekseyev, Aug 18 2013
a(51)-a(52) from Max Alekseyev, Jul 10 2016

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

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}]

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A001606 Indices of prime Lucas numbers.

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A072381 Numbers m such that Fibonacci(m) is a semiprime.

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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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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.

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A290246 Prime numbers that are common indices to both prime Lucas and prime Wagstaff numbers.

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