A001605 -id:A001605 - cp's OEIS frontend

A134851 Number of primes between A001605(n) and A001605(n+1).

0, 1, 1, 1, 1, 1, 2, 1, 4, 1, 8, 9, 1, 39, 11, 1, 3, 10, 7, 1, 324, 208, 73, 442, 42, 498, 1122, 502, 508, 147, 1235, 2796, 2014, 2145, 1520, 4388, 15584, 2814, 11888, 274, 826, 24119, 8554, 16877, 24680, 11591, 11503, 63625, 22803, 6374, 92008, 115147, 79772, 157711, 3110

Offset: 1

Artur Jasinski, Nov 13 2007

nonn

Cf. A000045, A000720, A001605, A050937, A075737, A090819, A134787.

a = {}; k = {}; Do[If[PrimeQ[Fibonacci[n]], AppendTo[k, n]], {n, 1, 1000}]; Do[AppendTo[a, PrimePi[k[[n + 1]]] - PrimePi[k[[n]]]], {n, 1, 20}]; a

a(n) = primepi(A001605(n+1)) - primepi(A001605(n)). - Amiram Eldar, Sep 01 2019

More terms from Amiram Eldar, Sep 01 2019 and Sep 15 2024

A135724 Fibonacci numbers whose indices are prime Fibonacci numbers: a(n) = Fibonacci(A001605(n)).

Original entry on oeis.org

1, 2, 5, 233, 1779979416004714189, 2211236406303914545699412969744873993387956988653

Offset: 1

Artur Jasinski, Nov 26 2007

nonn

The only known prime numbers in this sequence are 2, 5 and 233.

The next term (a(6)) has 334 digits. - Harvey P. Dale, Apr 28 2023

Amiram Eldar, Table of n, a(n) for n = 1..7

Cf. A000045, A001605, A135723, A134787, A134788, A134789, A134790.

a = {}; b = {}; Do[If[PrimeQ[c = Fibonacci[n]], w = Fibonacci[Fibonacci[n]]; AppendTo[a, w]; AppendTo[b, n]], {n, 1, 31}]; a
Fibonacci[#]&/@Select[Fibonacci[Range[20]],PrimeQ] (* Harvey P. Dale, Apr 28 2023 *)

Corrected by Harvey P. Dale, Apr 28 2023

A172115 Partial sums of A001605.

Original entry on oeis.org

3, 7, 12, 19, 30, 43, 60, 83, 112, 155, 202, 285, 416, 553, 912, 1343, 1776, 2225, 2734, 3303, 3874, 6845, 11568, 16955, 26266, 35943, 50374, 75935, 106692, 142691, 180202, 231035, 312874, 417785, 547806, 695897, 897004, 1294383, 1728164, 2318205, 2911894, 3516605

Offset: 1

Jonathan Vos Post, Jan 25 2010

			a(1) = 3.
a(2) = 3 + 4 = 7.
a(3) = 3 + 4 + 5 = 12.

Amiram Eldar, Table of n, a(n) for n = 1..56 (calculated from the data at A001605)

Cf. A001605, A001578, A005478, A086597, A080345.

a(40) onwards from Amiram Eldar, Jul 22 2025

A005478 Prime Fibonacci numbers.

Original entry on oeis.org

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241

Offset: 1

N. J. A. Sloane

a(n) == 1 (mod 4) for n > 2. (Proof. Otherwise 3 < a(n) = F_k == 3 (mod 4). Then k == 4 (mod 6) (see A079343 and A161553) and so k is not prime. But k is prime since F_k is prime and k != 4 - see Caldwell.)
More generally, A190949(n) == 1 (mod 4). - N. J. A. Sloane
With the exception of 3, every term of this sequence has a prime index in the sequence of Fibonacci numbers (A000045); e.g., 5 is the fifth Fibonacci number, 13 is the seventh Fibonacci number, 89 the eleventh, etc. - Alonso del Arte, Aug 16 2013
Note: A001605 gives those indices. - Antti Karttunen, Aug 16 2013
The six known safe primes 2p + 1 such that p is a Fibonacci prime are in A263880; the values of p are in A155011. There are only two known Fibonacci primes p for which 2p - 1 is also prime, namely, p = 2 and 3. Is there a reason for this bias toward prime 2p + 1 over 2p - 1 among Fibonacci primes p? - Jonathan Sondow, Nov 04 2015

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 89, p. 32, Ellipses, Paris 2008.
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).

N. J. A. Sloane, Table of n, a(n) for n = 1..23
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
Chris Caldwell's The Top Twenty, Fibonacci Number.
Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
Ron Knott, Mathematics of the Fibonacci Series
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, Fibonacci Prime

Cf. A001605, A000045, A030426, A075736, A099000, A263880.
Subsequence of A178762.
Column k=1 of A303216.

Select[Fibonacci[Range[400]], PrimeQ] (* Alonso del Arte, Oct 13 2011 *)

je=[]; for(n=0,400, if(isprime(fibonacci(n)),je=concat(je,fibonacci(n)))); je

from itertools import islice
from sympy import isprime
def A005478_gen(): # generator of terms
    a, b = 1, 1
    while True:
        if isprime(b):
            yield b
        a, b = b, a+b
A005478_list = list(islice(A005478_gen(),10)) # Chai Wah Wu, Jun 25 2024

[i for i in fibonacci_xrange(0,10^80) if is_prime(i)] # Bruno Berselli, Jun 26 2014

a(n) = A000045(A001605(n)). A000040 INTERSECT A000045. - R. J. Mathar, Nov 01 2007

Sequence corrected by Enoch Haga, Feb 11 2000
One more term from Jason Earls, Jul 12 2001
Comment and proof added by Jonathan Sondow, May 24 2011

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

A090819 Primes p such that the p-th Fibonacci number is nonprime.

Original entry on oeis.org

2, 19, 31, 37, 41, 53, 59, 61, 67, 71, 73, 79, 89, 97, 101, 103, 107, 109, 113, 127, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349

Offset: 1

Cino Hilliard, Feb 11 2004

Is it true that a(n) ~ n log n? - Charles R Greathouse IV, Aug 15 2015

			Fibonacci(37) = 24157817 = 73*149*2221. [corrected by _Bobby Jacobs_, Sep 25 2017]

Cf. A000045, A001605, A134787.

Essentially the same as A038672.

Select[Prime[Range[62]], ! PrimeQ[Fibonacci[#]] &] (* Jayanta Basu, Jul 10 2013 *)

f(n) = forprime(x=2,n,p=fibonacci(x);if(!isprime(p),print1(x",")))

a(n) = prime(A134787(n)). - Amiram Eldar, Oct 25 2024

Definition corrected by Don Reble, Sep 04 2008

A134852 Number of distinct prime factors of the Fibonacci numbers in A050937.

Original entry on oeis.org

0, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 2, 2, 4, 2, 3, 2, 2, 2, 2, 3, 4, 2, 4, 4, 2, 2, 3, 3, 2, 2, 4, 2, 4, 4, 2, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 3, 4, 2, 4, 4, 4, 3, 2, 3, 5, 4, 2, 7, 5, 4, 3, 3, 2, 2, 4, 3, 4, 5, 5, 3, 5, 3, 2, 3, 4, 3, 4, 6, 3, 4, 3, 5, 3, 5, 6, 2

Offset: 1

Artur Jasinski, Nov 13 2007

nonn

Charles R Greathouse IV and Amiram Eldar, Table of n, a(n) for n = 1..202 (terms 1..185 from Charles R Greathouse IV)

Cf. A000045, A001221, A001605, A050937, A075737, A090819, A134787, A134851.

a = {}; k = {}; Do[If[ ! PrimeQ[Fibonacci[Prime[n]]], c = Length[FactorInteger[Fibonacci[Prime[n]]]]; AppendTo[k, c]], {n, 1, 50}]; k

forprime(p=2,99,t=omega(fibonacci(p)); if(t!=1,print1(t", "))) \\ Charles R Greathouse IV, Feb 03 2014

a(n) = A001221(A050937(n)). - R. J. Mathar, May 03 2008

Edited by R. J. Mathar, May 03 2008

a(38)-a(87) from Charles R Greathouse IV, Feb 03 2014

A060384 Number of decimal digits in n-th Fibonacci number.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Labos Elemer, Apr 03 2001

Harry J. Smith, Table of n, a(n) for n = 0..10000

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.

Cf. A261585.

a060384 = a055642 . a000045  -- Reinhard Zumkeller, Mar 09 2013

with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); # Emeric Deutsch, May 19 2007

Table[IntegerLength@ Fibonacci@ n, {n, 0, 84}] /. 0 -> 1 (* or *)
Table[Floor[n Log10@ GoldenRatio - Log10@ 5/2] + 1, {n, 0, 84}] /. 0 -> 1 (* Michael De Vlieger, Jul 04 2016 *)

print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = #Str(fibonacci(n)); \\ Michel Marcus, Jul 04 2016

a(n) = floor(n*log(tau)/log(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre, Oct 29 2002. [Corrected by Hans J. H. Tuenter, Jul 07 2025].
a(n) = floor(n*log_10(gold) - log_10(5)/2) + 1 for n >= 2, where gold is (1+sqrt(5))/2. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007
a(n) = A055642(A000045(n)). - Reinhard Zumkeller, Mar 09 2013

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A303215 A(n,k) is the n-th index of a Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 8, 4, 6, 9, 5, 20, 15, 10, 7, 18, 27, 16, 14, 11, 12, 44, 28, 21, 19, 13, 30, 40, 45, 32, 25, 22, 17, 54, 42, 50, 57, 52, 33, 26, 23, 24, 78, 56, 64, 63, 55, 35, 31, 29, 36, 80, 102, 66, 75, 68, 74, 37, 34, 43, 138, 100, 88, 128, 70, 92, 69, 77, 38, 41, 47

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   3,  8,  6, 20, 18,  12,  30,  54,  24,  36, ...
   4,  9, 15, 27, 44,  40,  42,  78,  80, 100, ...
   5, 10, 16, 28, 45,  50,  56, 102,  88, 114, ...
   7, 14, 21, 32, 57,  64,  66, 128, 110, 165, ...
  11, 19, 25, 52, 63,  75,  70, 130, 112, 174, ...
  13, 22, 33, 55, 68,  92,  81, 135, 184, 256, ...
  17, 26, 35, 74, 69,  95, 104, 147, 186, 266, ...
  23, 31, 37, 77, 76,  99, 105, 154, 189, 273, ...
  29, 34, 38, 85, 91, 116, 136, 170, 196, 282, ...
  43, 41, 39, 87, 98, 117, 148, 171, 225, 296, ...

Alois P. Heinz, Antidiagonals n = 1..16, flattened

Columns k=1-13 give: A001605, A072381, A114812, A114813, A114814, A114815, A114816, A114817, A114818, A114819, A114820, A114821, A114822.
Row n=1 gives A072396.
Cf. A000045, A001222, A038575, A303216, A303217.

F:= combinat[fibonacci]: with(numtheory):
A:= proc() local h, p, q; p, q:= proc() [] end, 2;
      proc(n, k)
        while nops(p(k))

A[n_, k_] := Module[{h, p, q = 2}, p[k] = {}; While[Length[p[k]]Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *)

A000045(A(n,k)) = A303216(n,k).

A001222(A000045(A(n,k))) = k.

A060319 Smallest Fibonacci number with n distinct prime factors.

Original entry on oeis.org

1, 2, 21, 610, 6765, 832040, 102334155, 190392490709135, 1548008755920, 23416728348467685, 2880067194370816120, 81055900096023504197206408605, 2706074082469569338358691163510069157, 5358359254990966640871840, 57602132235424755886206198685365216, 18547707689471986212190138521399707760

Offset: 0

Labos Elemer, Mar 28 2001

nonn

			a(5) = F(30) = 832040 = 2^3 * 5 * 11 * 41 * 61.

Amiram Eldar, Table of n, a(n) for n = 0..40
Ron Knott, Fibonacci numbers with tables of F(0)-F(500)

Cf. A001605, A005478, A060320.

Row n=1 of A303218.

f[n_]:=Length@FactorInteger[Fibonacci[n]]; lst={};Do[Do[If[f[n]==q,Print[Fibonacci[n]];AppendTo[lst,Fibonacci[n]];Break[]],{n,280}],{q,18}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
First /@ SortBy[#, Last] &@ Map[#[[1]] &, Values@ GroupBy[#, Last]] &@ Table[{#, PrimeNu@ #} &@ Fibonacci@ n, {n, 2, 200}] (* Michael De Vlieger, Feb 18 2017, Version 10 *)

cp's OEIS Frontend

A134851 Number of primes between A001605(n) and A001605(n+1).

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A135724 Fibonacci numbers whose indices are prime Fibonacci numbers: a(n) = Fibonacci(A001605(n)).

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A172115 Partial sums of A001605.

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A005478 Prime Fibonacci numbers.

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A001606 Indices of prime Lucas numbers.

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A090819 Primes p such that the p-th Fibonacci number is nonprime.

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A134852 Number of distinct prime factors of the Fibonacci numbers in A050937.

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A060384 Number of decimal digits in n-th Fibonacci number.

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