A303215 -id:A303215 - cp's OEIS frontend

A001605 Indices of prime Fibonacci numbers.

3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433, 449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107, 397379, 433781, 590041, 593689, 604711, 931517, 1049897, 1285607, 1636007, 1803059, 1968721, 2904353, 3244369, 3340367

Offset: 1

N. J. A. Sloane

Some of the larger entries may only correspond to probable primes.
Since F(n) divides F(mn) (cf. A001578, A086597), all terms of this sequence are primes except for a(2) = 4 = 2 * 2 but F(2) = 1. - M. F. Hasler, Dec 12 2007
What is the next larger twin prime after F(4) = 3, F(5) = 5, F(7) = 13? The next candidates seem to be F(104911) or F(1968721) (greater of a pair), or F(397379), F(931517) (lesser of a pair). - M. F. Hasler, Jan 30 2013, edited Dec 24 2016, edited Sep 23 2017 by Bobby Jacobs
_Henri Lifchitz_ confirms that the data section gives the full list (49 terms) as far as we know it today of indices of prime Fibonacci numbers (including proven primes and PRPs). - N. J. A. Sloane, Jul 09 2016
Terms n such that n-2 is also a term are listed in A279795. - M. F. Hasler, Dec 24 2016
There are no Fibonacci numbers that are twin primes after F(7) = 13. Every Fibonacci prime greater than F(4) = 3 is of the form F(2*n+1). Since F(2*n+1)+2 and F(2*n+1)-2 are F(n+2)*L(n-1) and F(n-1)*L(n+2) in some order, and F(n+2) > 1, L(n-1) > 1, F(n-1) > 1, and L(n+2) > 1 for n > 3. - Bobby Jacobs, Sep 23 2017
These primes are occurring with about the same normalized frequency as Repunit primes (see Generalized Repunit Conjecture Ref).  Assuming a base=1.618 (ratio of sequential terms), then the best fit coefficient is 0.60324 for the first 56 terms, which is already approaching Euler's constant 0.56145948. - Paul Bourdelais, Aug 23 2024

Clifford A. Pickover, Mazes for the Mind, St. Martin's Press, NY, 1992, p. 350.
Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 54.
Paulo Ribenboim, The Little Book of Big Primes, Springer-Verlag, NY, 1991, p. 178.
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).

Paul Bourdelais, Table of n, a(n) for n = 1..56 (first 51 terms from Henri Lifchitz)
Paul Bourdelais, A Generalized Repunit Conjecture
John Brillhart, Peter L. Montgomery and Robert D. Silverman, Tables of Fibonacci and Lucas factorizations, Math. Comp. 50 (1988), 251-260.
David Broadhurst, Fibonacci Numbers
David Broadhurst, Proof that F(81839) is prime, NMBRTHRY Mailing List, Apr 22 2001.
Chris K. Caldwell, The Prime Glossary, Fibonacci prime
Rosina Campbell, Duc Van Huynh, Tyler Melton, and Andrew Percival, Elliptic Curves of Fibonacci order over F_p, arXiv:1710.05687 [math.NT], 2017.
Harvey Dubner and Wilfrid Keller, New Fibonacci and Lucas Primes, Math. Comp. 68 (1999) 417-427.
Dudley Fox, Search for Possible Fibonacci Primes
Shyam Sunder Gupta, Fabulous Fibonacci Numbers, Lucas Numbers, and Golden Ratio, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 8, 223-274.
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See p. 36.
Ron Knott, Mathematics of the Fibonacci Series
Alex Kontorovich and Jeff Lagarias, On Toric Orbits in the Affine Sieve, arXiv:1808.03235 [math.NT], 2018.
Henri & Renaud Lifchitz, PRP Records.
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.
Rudolf Ondrejka, The Top Ten: a Catalogue of Primal Configurations
PRP Top Records, Search for: F(n)
Lawrence Somer and Michal Křížek, On Primes in Lucas Sequences, Fibonacci Quart. 53 (2015), no. 1, 2-23.
Eric Weisstein's World of Mathematics, Fibonacci Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000045, A001578, A005478, A080345, A086597, A117595.
Subsequence of A046022.
Column k=1 of A303215.

Select[Range[10^4], PrimeQ[Fibonacci[#]] &] (* Harvey P. Dale, Nov 20 2012 *)
(* Start ~ 1.8x faster than the above *)
Select[Range[10^4], PrimeQ[#] && PrimeQ[Fibonacci[#]] &] (* Eric W. Weisstein, Nov 07 2017 *)
Select[Prime[Range[PrimePi[10^4]]], PrimeQ[Fibonacci[#]] &] (* Eric W. Weisstein, Nov 07 2017 *)
(* End *)

v=[3,4]; forprime(p=5,1e5, if(ispseudoprime(fibonacci(p)), v=concat(v,p))); v \\ Charles R Greathouse IV, Feb 14 2011

is_A001605(n)={n==4 || isprime(n) & ispseudoprime(fibonacci(n))}  \\ M. F. Hasler, Sep 29 2012

Prime(i) = a(n) for some n <=> A080345(i) <= 1. - M. F. Hasler, Dec 12 2007

Additional comments from Robert G. Wilson v, Aug 18 2000
More terms from David Broadhurst, Nov 08 2001
Two more terms (148091 and 201107) from T. D. Noe, Feb 12 2003 and Mar 04 2003
397379 from T. D. Noe, Aug 18 2003
433781, 590041, 593689 from Henri Lifchitz submitted by Ray Chandler, Feb 11 2005
604711 from Henri Lifchitz communicated by Eric W. Weisstein, Nov 29 2005
931517, 1049897, 1285607 found by Henri Lifchitz circa Nov 01 2008 and submitted by Alexander Adamchuk, Nov 28 2008
1636007 from Henri Lifchitz March 2009, communicated by Eric W. Weisstein, Apr 24 2009
1803059 and 1968721 from Henri Lifchitz, November 2009, submitted by Alex Ratushnyak, Aug 08 2012
a(49)=2904353 from Henri Lifchitz, Jul 15 2014
a(50)=3244369 from Henri Lifchitz, Nov 04 2017
a(51)=3340367 from Henri Lifchitz, Apr 25 2018
a(52)-a(56) from Ryan Propper added by Paul Bourdelais, Aug 23 2024

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

A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

3, 8, 4, 15, 9, 5, 20, 16, 10, 6, 30, 24, 18, 12, 7, 40, 36, 27, 21, 14, 11, 70, 48, 42, 28, 33, 19, 13, 60, 81, 54, 44, 32, 35, 22, 17, 80, 72, 104, 56, 45, 52, 37, 25, 23, 90, 84, 110, 105, 64, 50, 55, 38, 26, 29, 140, 126, 88, 112, 136, 78, 57, 74, 39, 31, 43

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   3,  8, 15, 20, 30,  40,  70,  60,  80,  90, ...
   4,  9, 16, 24, 36,  48,  81,  72,  84, 126, ...
   5, 10, 18, 27, 42,  54, 104, 110,  88, 165, ...
   6, 12, 21, 28, 44,  56, 105, 112,  96, 256, ...
   7, 14, 33, 32, 45,  64, 136, 114, 100, 258, ...
  11, 19, 35, 52, 50,  78, 148, 128, 108, 266, ...
  13, 22, 37, 55, 57,  92, 152, 130, 132, 296, ...
  17, 25, 38, 74, 63,  95, 164, 135, 138, 304, ...
  23, 26, 39, 77, 66,  99, 182, 147, 156, 322, ...
  29, 31, 46, 85, 68, 102, 186, 154, 184, 369, ...

Alois P. Heinz, Antidiagonals n = 1..18, flattened

Columns k=2-16 give: A114842, A114841, A114843, A114840, A114839, A114838, A114837, A114836, A114826, A114825, A114824, A114823, A117529, A117551, A117550.
Row n=1 gives: A060320.
Cf. A000045, A001221, A022307, A303215, A303218.

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

nmax = 12; maxIndex = 200;
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k&];
T = Array[col, nmax];
A[n_, k_] := T[[k, n]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 04 2020 *)

A000045(A(n,k)) = A303218(n,k).

A001221(A000045(A(n,k))) = k.

A303216 A(n,k) is the n-th 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

2, 21, 3, 8, 34, 5, 6765, 610, 55, 13, 2584, 196418, 987, 377, 89, 144, 701408733, 317811, 10946, 4181, 233, 832040, 102334155, 1134903170, 2178309, 75025, 17711, 1597, 86267571272, 267914296, 12586269025, 365435296162, 32951280099, 3524578, 121393, 28657

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
    2,    21,       8,         6765,           2584,                 144, ...
    3,    34,     610,       196418,      701408733,           102334155, ...
    5,    55,     987,       317811,     1134903170,         12586269025, ...
   13,   377,   10946,      2178309,   365435296162,      10610209857723, ...
   89,  4181,   75025,  32951280099,  6557470319842,    2111485077978050, ...
  233, 17711, 3524578, 139583862445, 72723460248141, 7540113804746346429, ...

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

Columns k=1-2 give: A005478, A053409.
Row n=1 gives A072397.
Cf. A000045, A001222, A303215, A303218.

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[{F = Fibonacci, h, p, q = 2}, p[_] = {}; While[ Length[p[k]] < n, q = q+1; h = PrimeOmega[F[q]]; p[h] = Append[p[h], F[q]]]; p[k][[n]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 10}] // Flatten (* Jean-François Alcover, Feb 05 2021, after Alois P. Heinz *)

A(n,k) = A000045(A303215(n,k)).

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

A072396 Index of smallest Fibonacci number with n prime factors when counted with multiplicity.

Original entry on oeis.org

3, 8, 6, 20, 18, 12, 30, 54, 24, 36, 138, 48, 84, 72, 108, 96, 210, 120, 276, 168, 216, 252, 288, 240, 336, 570, 384, 420, 360, 576, 480, 540, 504, 660, 600, 672, 990, 720, 792, 840, 1152, 1140

Offset: 1

Shyam Sunder Gupta, Jul 21 2002

1452 < a(43) <= 1596, a(44) = 1296, a(45) = 1368, a(46) = 1080, a(47) = 1200, a(48) <= 1728. - Daniel Suteu, Jan 19 2023

			a(3) = 6 since the 6th Fibonacci number 8 has 3 prime factors.

Hisanori Mishima, Fibonacci numbers (n = 1 to 100).
Hisanori Mishima, Fibonacci numbers (n = 101 to 200).
Hisanori Mishima, Fibonacci numbers (n = 201 to 300).
Hisanori Mishima, Fibonacci numbers (n = 301 to 400).
Hisanori Mishima, Fibonacci numbers (n = 401 to 480).

Cf. A000045, A038575, A072397.

Row n=1 of A303215.

a(n) = my(k=1); while (bigomega(fibonacci(k)) != n, k++); k; \\ Michel Marcus, Aug 26 2020

a(17)-a(24) from Alois P. Heinz, Apr 10 2018

a(25)-a(42) from Amiram Eldar, Aug 26 2020

A114813 Indices of Fibonacci numbers with 4 prime factors when counted with multiplicity.

Original entry on oeis.org

20, 27, 28, 32, 52, 55, 74, 77, 85, 87, 93, 97, 115, 123, 143, 146, 149, 157, 161, 163, 178, 187, 197, 209, 211, 214, 215, 221, 223, 239, 242, 249, 262, 269, 283, 287, 307, 311, 313, 321, 334, 349, 379, 391, 393, 409, 421, 453, 487, 493, 499, 523, 581, 586

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

1559, 1609, 2131, 2281, 2351, 2459, 2539, 3119, 3371, 4993, 5839, 6217, 7591, 7741, 8353, 9931 are also terms (data from Kelly link). - Chai Wah Wu, Nov 11 2019

			a(1)=20 because 20th Fibonacci number (i.e., 6765) consists of 4 prime factors (i.e., 3*5*11*41).

Amiram Eldar, Table of n, a(n) for n = 1..86
Blair Kelly, Fibonacci and Lucas Factorizations.

Column k=4 of A303215.

t = {}; Do[f = FactorInteger[Fibonacci[n]]; If[Total[Transpose[f][[2]]] == 4, AppendTo[t, n]], {n, 2, 100}]; t (* T. D. Noe, Mar 14 2014 *)

n=1;while(n<340,if(bigomega(fibonacci(n))==4,print1(n,", "));n++)

More terms from Ryan Propper, May 22 2006

A114812 Indices of Fibonacci numbers with 3 prime factors when counted with multiplicity.

Original entry on oeis.org

6, 15, 16, 21, 25, 33, 35, 37, 38, 39, 46, 49, 51, 58, 62, 65, 67, 82, 86, 103, 106, 119, 122, 139, 142, 145, 158, 166, 179, 181, 226, 233, 235, 241, 257, 263, 274, 281, 299, 301, 317, 337, 383, 389, 419, 457, 463, 473, 479, 491, 521, 541, 557, 619, 643, 659

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

1811, 1933, 1997, 2069, 2087, 2203, 2221, 2311, 2663, 2713, 3631, 4157, 4651, 5107, 6701, 7211, 8123 are also terms (from data in Kelly link). - Chai Wah Wu, Nov 11 2019

			a(2)=15 because 15th Fibonacci number (i.e., 610) consists of 3 prime factors (i.e., 2*5*61).

Amiram Eldar, Table of n, a(n) for n = 1..83
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A072381, A114813.

Column k=3 of A303215.

t = {}; Do[f = FactorInteger[Fibonacci[n]]; If[Total[Transpose[f][[2]]] == 3, AppendTo[t, n]], {n, 2, 100}]; t (* T. D. Noe, Mar 14 2014 *)
Flatten[Position[Fibonacci[Range[700]],?(PrimeOmega[#]==3&)]] (* _Harvey P. Dale, Feb 15 2015 *)

n=1;while(n<340,if(bigomega(fibonacci(n))==3,print1(n,", "));n++)

{n: A038575(n)=3}. [R. J. Mathar, Jun 08 2010]

More terms from Ryan Propper, May 22 2006

A114814 Indices of Fibonacci numbers with 5 prime factors when counted with multiplicity.

Original entry on oeis.org

18, 44, 45, 57, 63, 68, 69, 76, 91, 98, 111, 118, 124, 125, 134, 141, 169, 172, 183, 185, 201, 202, 203, 213, 218, 229, 247, 253, 267, 302, 303, 329, 335, 347, 363, 371, 373, 377, 381, 382, 386, 395, 398, 413, 415, 439, 443, 461, 497, 501, 529, 547, 563, 579

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=18 because 18th Fibonacci number (i.e., 2584) consists of 5 prime factors (i.e., 2*2*2*17*19).

Amiram Eldar, Table of n, a(n) for n = 1..112
Blair Kelly, Fibonacci and Lucas Factorizations.

Column k=5 of A303215.

t = {}; Do[f = FactorInteger[Fibonacci[n]]; If[Total[Transpose[f][[2]]] == 5, AppendTo[t, n]], {n, 2, 100}]; t (* T. D. Noe, Mar 14 2014 *)

n=1;while(n<385,if(bigomega(fibonacci(n))==5,print1(n,", "));n++)

More terms from Ryan Propper, May 22 2006

A114815 Indices of Fibonacci numbers with 6 prime factors when counted with multiplicity.

Original entry on oeis.org

12, 40, 50, 64, 75, 92, 95, 99, 116, 117, 129, 133, 153, 155, 159, 177, 188, 194, 205, 206, 219, 237, 245, 265, 278, 314, 323, 327, 339, 341, 343, 346, 356, 358, 361, 362, 394, 407, 411, 417, 422, 427, 437, 446, 454, 466, 482, 502, 503, 505, 514, 515, 527

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1)=12 because 12th Fibonacci number (i.e., 144) consists of 6 prime factors (i.e., 2*2*2*2*3*3).

Amiram Eldar, Table of n, a(n) for n = 1..120
Blair Kelly, Fibonacci and Lucas Factorizations.

Column k=6 of A303215.

t = {}; Do[f = FactorInteger[Fibonacci[n]]; If[Total[Transpose[f][[2]]] == 6, AppendTo[t, n]], {n, 2, 100}]; t (* T. D. Noe, Mar 14 2014 *)
Position[Fibonacci[Range[550]],?(PrimeOmega[#]==6&)]//Flatten (* _Harvey P. Dale, Jun 12 2017 *)

n=1;while(n<330,if(bigomega(fibonacci(n))==6,print1(n,", "));n++)

More terms from Ryan Propper, May 22 2006

A114816 Indices of Fibonacci numbers with 7 prime factors when counted with multiplicity.

Original entry on oeis.org

30, 42, 56, 66, 70, 81, 104, 105, 136, 148, 152, 164, 175, 195, 207, 212, 244, 254, 259, 289, 291, 292, 298, 305, 319, 326, 332, 344, 365, 367, 403, 404, 423, 445, 447, 451, 458, 478, 489, 511, 517, 519, 526, 533, 537, 543, 554, 565, 566, 597, 605, 679, 681

Offset: 1

Shyam Sunder Gupta, Feb 19 2006

nonn

			a(1) = 30 because 30th Fibonacci number (i.e., 832040) consists of 7 prime factors (i.e., 2*2*2*5*11*31*61).

Amiram Eldar, Table of n, a(n) for n = 1..112
Blair Kelly, Fibonacci and Lucas Factorizations.

Cf. A000045.

Column k=7 of A303215.

t = {}; Do[f = FactorInteger[Fibonacci[n]]; If[Total[Transpose[f][[2]]] == 7, AppendTo[t, n]], {n, 2, 100}]; t (* T. D. Noe, Mar 14 2014 *)

n=1;while(n<310,if(bigomega(fibonacci(n))==7,print1(n,", "));n++)

More terms from Ryan Propper, May 22 2006

cp's OEIS Frontend

A001605 Indices of prime Fibonacci numbers.

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

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A303217 A(n,k) is the n-th index of a Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A303216 A(n,k) is the n-th Fibonacci number with exactly k prime factors (counted with multiplicity); square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A072396 Index of smallest Fibonacci number with n prime factors when counted with multiplicity.

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A114813 Indices of Fibonacci numbers with 4 prime factors when counted with multiplicity.

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A114812 Indices of Fibonacci numbers with 3 prime factors when counted with multiplicity.

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