A303216 -id:A303216 - cp's OEIS frontend

A005478 Prime Fibonacci numbers.

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

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.

A053409 Fibonacci numbers which are semiprimes.

Original entry on oeis.org

21, 34, 55, 377, 4181, 17711, 121393, 1346269, 5702887, 165580141, 53316291173, 956722026041, 2504730781961, 308061521170129, 806515533049393, 14472334024676221, 1779979416004714189, 19740274219868223167, 573147844013817084101, 10284720757613717413913

Offset: 1

G. L. Honaker, Jr., Jan 09 2000

nonn

Subsequence of A006881, since the only square Fibonacci numbers are 1 and 144. - Charles R Greathouse IV, Sep 24 2012

Apart from a(1) = 21, all terms are of the form F(p), F(2p), or F(p^2) where F(n) is the n-th Fibonacci number. - Charles R Greathouse IV, Oct 06 2016

Vincenzo Librandi, Table of n, a(n) for n = 1..48

Cf. A000045, A001358, A006881, A072381.

Column k=2 of A303216.

Select[Fibonacci@Range[120],Last/@FactorInteger[#]=={1,1}&] (* Vladimir Joseph Stephan Orlovsky, Jan 29 2012 *)
Select[Fibonacci[Range[150]],PrimeOmega[#]==2&] (* Harvey P. Dale, Jun 26 2020 *)

issemi(n)=bigomega(n)==2
list(lim)=my(v=List([21]),F,t); forprime(p=2,, F=fibonacci(p); if(F>lim, break); if(issemi(F), listput(v,F))); forprime(p=2,, F=fibonacci(p^2); if(F>lim, break); if(isprime(t=fibonacci(p)) && isprime(F/t), listput(v,F))); forprime(p=2,, F=fibonacci(2*p); if(F>lim, break); if(isprime(t=fibonacci(p)) && isprime(F/t), listput(v,F))); Set(v) \\ Charles R Greathouse IV, Oct 06 2016

a(n) = A000045(A072381(n)).

A072397 Smallest Fibonacci number with n prime factors when counted with multiplicity.

Original entry on oeis.org

2, 21, 8, 6765, 2584, 144, 832040, 86267571272, 46368, 14930352, 30960598847965113057878492344, 4807526976, 160500643816367088, 498454011879264, 16641027750620563662096, 51680708854858323072, 34507973060837282187130139035400899082304280

Offset: 1

Shyam Sunder Gupta, Jul 21 2002

nonn

			21 is the 2nd term because 21 is the smallest Fibonacci number having 2 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, A072396.

Row n=1 of A303216.

a(n) = A000045(A072396(n)). - Alois P. Heinz, Apr 10 2018

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

A303218 A(n,k) is the n-th 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

2, 21, 3, 610, 34, 5, 6765, 987, 55, 8, 832040, 46368, 2584, 144, 13, 102334155, 14930352, 196418, 10946, 377, 89, 190392490709135, 4807526976, 267914296, 317811, 3524578, 4181, 233, 1548008755920, 37889062373143906, 86267571272, 701408733, 2178309, 9227465, 17711, 1597

Offset: 1

Alois P. Heinz, Apr 19 2018

			Square array A(n,k) begins:
   2,   21,     610,        6765,      832040,        102334155, ...
   3,   34,     987,       46368,    14930352,       4807526976, ...
   5,   55,    2584,      196418,   267914296,      86267571272, ...
   8,  144,   10946,      317811,   701408733,     225851433717, ...
  13,  377, 3524578,     2178309,  1134903170,   10610209857723, ...
  89, 4181, 9227465, 32951280099, 12586269025, 8944394323791464, ...

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

Column k=3 gives A137563.
Row n=1 gives: A060319.
Cf. A000045, A001221, 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))

nmax = 12(*rows*);
maxIndex = 200; (* increase if message "part does not exist" *)
nu[n_] := nu[n] = PrimeNu[Fibonacci[n]];
col[k_] := Select[Range[maxIndex], nu[#] == k &];
T = Array[col, nmax];
A[n_, k_] := Fibonacci[T[[k, n]]];
Table[A[n-k+1, k], {n, 1, nmax}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Feb 05 2021 *)

A(n,k) = A000045(A303217(n,k)).

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

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A005478 Prime Fibonacci numbers.

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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.

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A053409 Fibonacci numbers which are semiprimes.

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A072397 Smallest Fibonacci number with n prime factors when counted with multiplicity.

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A303218 A(n,k) is the n-th Fibonacci number with exactly k distinct prime factors; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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