A086597 -id:A086597 - 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

A001602 Fibonacci entry points: a(n) = smallest m > 0 such that the n-th prime divides Fibonacci(m).

Original entry on oeis.org

3, 4, 5, 8, 10, 7, 9, 18, 24, 14, 30, 19, 20, 44, 16, 27, 58, 15, 68, 70, 37, 78, 84, 11, 49, 50, 104, 36, 27, 19, 128, 130, 69, 46, 37, 50, 79, 164, 168, 87, 178, 90, 190, 97, 99, 22, 42, 224, 228, 114, 13, 238, 120, 250, 129, 88, 67, 270, 139, 28, 284, 147, 44, 310

Offset: 1

N. J. A. Sloane

"[a(n)] is called by Lucas the rank of apparition of p and we know it is a divisor of, or equal to prime(n)-1 or prime(n)+1" - Vajda, p. 84. (Note that a(3)=5. This is the only exception.) - Chris K. Caldwell, Nov 03 2008
Every number except 1, 2, 6 and 12 eventually occurs in this sequence. See also A086597(n), the number of primitive prime factors of Fibonacci(n). - T. D. Noe, Jun 13 2008
For each prime p we have an infinite sequence of integers, F(i*a(n))/p, i=1,2,... See also A236479. For primes p >= 3 and exponents j >= 2, with k = a(n) and p = p(n), it appears that F(k*i*p^(j-1))/p^j is an integer, for i >= 0. For p = 2, F(k*i*p^(j-1))/p^(j+1) = integer. - Richard R. Forberg, Jan 26-29 2014 [Comments revised by N. J. A. Sloane, Sep 24 2015]
Let p=prime(n). a(n) is also a divisor of (p-1)/2 (if p mod 5 == 1 or 4) or (p+1)/2 (if p mod 5 == 2 or 3) if and only if p mod 4 = 1. - Seiichi Azuma, Jul 29 2014

			The 5th prime is 11 and 11 first divides Fib(10)=55, so a(5) = 10.

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, Ellis Horwood Ltd., Chichester, 1989.

T. D. Noe, Table of n, a(n) for n = 1..10000
U. Alfred, M. Wunderlich, Tables of Fibonacci Entry Points, Part I, (1965).
Miho Aoki and Yuho Sakai, On Equivalence Classes of Generalized Fibonacci Sequences, JIS vol 19 (2016) # 16.2.6
B. Avila, T. Khovanova, Free Fibonacci SequencesJ. Int. Seq. 17 (2014) # 14.8.5.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, Fibonacci Association, San Jose, CA, 1972. See p. 25.
Paul Cubre and Jeremy Rouse, Divisibility properties of the Fibonacci entry point, arXiv:1212.6221 [math.NT], 2012.
D. E. Daykin and L. A. G. Dresel, Factorization of Fibonacci Numbers part 2, Fibonacci Quarterly, vol 8 (1970), pages 23 - 30 and 82.
Ramon Glez-Regueral, An entry-point algorithm for high-speed factorization, Thirteenth Internat. Conf. Fibonacci Numbers Applications, Patras, Greece, 2008.
R. K. Guy, The Second Strong Law of Small Numbers, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy, pp. 2-3 missing] See p. 7.
D. Lind et al., Tables of Fibonacci entry points, part 2, reviewed in, Math. Comp., 20 (1966), 618-619.
Patrick McKinley, Table of a(n) for n=1..678921
Daniel Yaqubi, Amirali Fatehizadeh, Some results on average of Fibonacci and Lucas sequences, arXiv:2001.11839 [math.CO], 2020.

Cf. A051694, A001177, A086597, A194363 (entries Lucas).

import Data.List (findIndex)
import Data.Maybe (fromJust)
a001602 n = (+ 1) $ fromJust $
            findIndex ((== 0) . (`mod` a000040 n)) $ tail a000045_list
-- Reinhard Zumkeller, Apr 08 2012

A001602 := proc(n)
    local i,p;
    p := ithprime(n);
    for i from 1 do
        if modp(combinat[fibonacci](i),p) = 0 then
            return i;
        end if;
    end do:
end proc: # R. J. Mathar, Oct 31 2015

Table[k=1;While[!Divisible[Fibonacci[k],Prime[n]],k++];k,{n,70}] (* Harvey P. Dale, Feb 15 2012 *)
(* a fast, but more complicated method *) MatrixPowerMod[mat_, n_, m_Integer] := Mod[Fold[Mod[If[#2 == 1, #1.#1.mat, #1.#1], m] &, mat, Rest[IntegerDigits[n, 2]]], m]; FibMatrix[n_Integer, m_Integer] := MatrixPowerMod[{{0, 1}, {1, 1}}, n, m]; FibEntryPointPrime[p_Integer] := Module[{n, d, k}, If[PrimeQ[p], n = p - JacobiSymbol[p, 5]; d = Divisors[n]; k = 1; While[FibMatrix[d[[k]], p][[1, 2]] > 0, k++]; d[[k]]]]; SetAttributes[FibEntryPointPrime, Listable]; FibEntryPointPrime[Prime[Range[10000]]] (* T. D. Noe, Jan 03 2013 *)
With[{nn=70,t=Table[{n,Fibonacci[n]},{n,500}]},Transpose[ Flatten[ Table[ Select[t,Divisible[#[[2]],Prime[i]]&,1],{i,nn}],1]][[1]]] (* Harvey P. Dale, May 31 2014 *)

a(n)=if(n==3,5,my(p=prime(n));fordiv(p^2-1,d,if(fibonacci(d)%p==0, return(d)))) \\ Charles R Greathouse IV, Jul 17 2012

do(p)=my(k=p+[0,-1,1,1,-1][p%5+1],f=factor(k));for(i=1,#f[,1],for(j=1,f[i,2],if((Mod([1,1;1,0],p)^(k/f[i,1]))[1,2], break); k/=f[i,1])); k
a(n)=do(prime(n))
apply(do, primes(100)) \\ Charles R Greathouse IV, Jan 03 2013

from sympy.ntheory.generate import prime
def A001602(n):
    a, b, i, p = 0, 1, 1, prime(n)
    while b % p:
        a, b, i = b, (a+b) % p, i+1
    return i # Chai Wah Wu, Nov 03 2015, revised Apr 04 2016.

a(n) = A001177(prime(n)).

a(n) <= prime(n) + 1. - Charles R Greathouse IV, Jan 02 2013

More terms from Jud McCranie

A022307 Number of distinct prime factors of n-th Fibonacci number.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 2, 3, 3, 1, 3, 2, 4, 3, 2, 1, 4, 2, 2, 4, 4, 1, 5, 2, 4, 3, 2, 3, 5, 3, 3, 3, 6, 2, 5, 1, 5, 5, 3, 1, 6, 3, 5, 3, 4, 2, 6, 4, 6, 5, 3, 2, 8, 2, 3, 5, 6, 3, 5, 3, 5, 5, 7, 2, 8, 2, 4, 5, 5, 4, 6, 2, 9, 7, 3, 1, 9, 4, 3, 4, 9, 2, 10, 4, 6, 4, 2, 6, 9, 4, 5, 6

Offset: 0

Clark Kimberling

nonn

Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. Exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. - Jonathan Vos Post, Dec 06 2006
First occurrence of k: 0, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, ..., . - Robert G. Wilson v, Dec 10 2006 [Other than 0, this is sequence A060320. - Jon E. Schoenfield, Dec 30 2016]
Row lengths of table A060442. - Reinhard Zumkeller, Aug 30 2014
If k properly divides n then a(n) >= a(k) + 1, except for a(6) = a(3) = 1. - Robert Israel, Aug 18 2015

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.

Max Alekseyev, Table of n, a(n) for n = 0..1422 (terms 0..1000 and 1001..1408 from T. D. Noe and Amiram Eldar, respectively).
Blair Kelly, Fibonacci and Lucas Factorizations
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449-461.
J. C. Lagarias, Errata to: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 162, No. 2, (1994), 393-396.
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A038575 (number of prime factors, counting multiplicity), A086597 (number of primitive prime factors).
Cf. A000032, A000040, A000045, A001221, A053028.
Cf. A060442, A086598 (omega(Lucas(n))).
Cf. A060320. - Jon E. Schoenfield, Dec 30 2016

a022307 n = if n == 0 then 0 else a001221 $ a000045 n
-- Reinhard Zumkeller, Aug 30 2014

[0] cat [#PrimeDivisors(Fibonacci(n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Table[Length[FactorInteger[Fibonacci[n]]], {n, 150}]

a(n)=omega(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014

a(n) = Sum{d|n} A086597(d), Mobius transform of A086597.

a(n) = A001221(A000045(n)) = omega(F(n)). - Jonathan Vos Post, Dec 06 2006

A038575 Number of prime factors of n-th Fibonacci number, counted with multiplicity.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 3, 1, 2, 2, 2, 1, 6, 1, 2, 3, 3, 1, 5, 2, 4, 3, 2, 1, 9, 3, 2, 4, 4, 1, 7, 2, 4, 3, 2, 3, 10, 3, 3, 3, 6, 2, 7, 1, 5, 5, 3, 1, 12, 3, 6, 3, 4, 2, 8, 4, 7, 5, 3, 2, 12, 2, 3, 5, 6, 3, 7, 3, 5, 5, 7, 2, 14, 2, 4, 6, 5, 4, 8, 2, 9, 7, 3, 1, 13, 4, 3, 4, 9, 2, 12, 5, 6, 4, 2, 6, 16, 4, 5, 6, 10, 2, 8

Offset: 0

Jeff Burch

nonn

Row lengths of table A060441. - Reinhard Zumkeller, Aug 30 2014

			a(12) = 6 because Fibonacci(12) = 144 = 2^4 * 3^2 has 6 prime factors.

Amiram Eldar, Table of n, a(n) for n = 0..1408 (terms 0..1000 from T. D. Noe derived from Kelly's data)
Blair Kelly, Fibonacci and Lucas Factorizations.
Douglas Lind, Problem H-145, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 351; Factor Analysis, Solution to Problem H-145 by the proposer, ibid., Vol. 8, No. 4 (1970), pp. 386-387.
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A022307 (number of distinct prime factors), A086597 (number of primitive prime factors).

Cf. also A001222, A000045, A060441.

a038575 n = if n == 0 then 0 else a001222 $ a000045 n
-- Reinhard Zumkeller, Aug 30 2014

with(numtheory):with(combinat):a:=proc(n) if n=0 then 0 else bigomega(fibonacci(n)) fi end: seq(a(n), n=0..102); # Zerinvary Lajos, Apr 11 2008

Join[{0, 0}, Table[Plus@@(Transpose[FactorInteger[Fibonacci[n]]][[2]]), {n, 3, 102}]]
Join[{0},PrimeOmega[Fibonacci[Range[110]]]] (* Harvey P. Dale, Apr 14 2018 *)

a(n)=bigomega(fibonacci(n)) \\ Charles R Greathouse IV, Sep 14 2015

from sympy import primeomega, fibonacci
def a(n): return 0 if n == 0 else primeomega(fibonacci(n))
print([a(n) for n in range(103)]) # Michael S. Branicky, Feb 02 2022

For n > 0: a(n) = A001222(A000045(n)). - Reinhard Zumkeller, Aug 30 2014

a(n) >= A001222(n) - 1 (Lind, 1968). - Amiram Eldar, Feb 02 2022

A001578 Smallest primitive prime factor of Fibonacci number F(n), or 1 if F(n) has no primitive prime factor.

Original entry on oeis.org

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, 139, 2971215073, 1103, 97, 101

Offset: 1

N. J. A. Sloane

nonn

A prime factor of F(n) is called primitive if it does not divide F(r) for any r < n.
A Fibonacci number can have more than one primitive factor; the primitive factors of F(19) are 37 and 113.
From Robert Israel, Oct 13 2015: (Start)
Since gcd(F(n),F(k)) = F(gcd(n,k)), the non-primitive prime factors of F(n) are factors of F(k) for some proper divisors k of n.
Since prime p divides F(p-1) if p == 1 or 4 (mod 5), F(p+1) if p == 2 or 3 mod 5, F(p) if p = 5, we have a(n) >= n-1 if a(n) > 1.
a(n) = n-1 iff n=2 or n-1 is in A000057.
a(n) = n+1 iff n+1 is a prime in A106535. (End)

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).

Max Alekseyev, Table of n, a(n) for n = 1..1452 (terms 1..1000 from Blair Kelly and T. D. Noe, terms 1409..1422 from Tyler Busby)
Mansur S. Boase, A Result About the Primes Dividing Fibonacci Numbers, The Fibonacci Quarterly, 39.5 (2001) 386.
R. D. Carmichael, On the numerical factors of the arithmetic forms α^n ± β^n, Annals of Math., 15 (1/4) (1913), 30-70.
D. Jarden, On the greatest primitive divisors of Fibonacci and Lucas numbers with prime-power subscripts, Fib. Quart. 1(#3) (1963), 15-31.
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000045, A000057, A106535, A086597 (number of primitive prime factors in F(n)), A061488 (1's omitted), A262341 (largest primitive prime factor of F(n)).

for n from 1 to 350 do
  f:= combinat:-fibonacci(n);
  if not isprime(n) then
    for k in map(t -> n/t, numtheory:-factorset(n)) do
       fk:= combinat:-fibonacci(k);
       g:= igcd(f,fk);
       while g > 1 do
         f:= f/g;
         g:= igcd(f,fk);
       od
    od
  fi;
  if f = 1 then A[n]:= 1; next fi;
  F:= map(t -> t[1],ifactors(f,easy)[2]);
  p:= select(type, F,integer);
  if nops(p) >= 1 then A[n]:= min(p); next fi;
  A[n]:= min(numtheory:-factorset(f));
od:
seq(A[i],i=1..350); # Robert Israel, Oct 13 2015

prms={}; Table[f=First/@FactorInteger[Fibonacci[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 50}]

a(n) = {my(v = vector(n, k, fibonacci(k))); my(vf = vector(n, k, factor(v[k])[,1]~)); for (k=1, n-1, vf[n] = setminus(vf[n], vf[k]);); if (#vf[n], vecmin(vf[n]), 1);} \\ Michel Marcus, May 11 2021

a(n) = 1 if and only if n = 1, 2, 6, or 12, by Carmichael's theorem. - Jonathan Sondow, Dec 07 2017

Edited by T. D. Noe, Apr 15 2004

Definition clarified at the suggestion of Joerg Arndt by Jonathan Sondow, Oct 13 2015

A152012 Indices of Fibonacci numbers having exactly one primitive prime factor.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 60, 62, 63, 65, 66, 72, 74, 75, 76, 82, 83, 93, 94, 98, 105, 106, 108, 111, 112, 119, 121, 122, 123, 124, 125, 131

Offset: 1

Max Alekseyev, Nov 19 2008

nonn

It is known that Fibonacci number A000045(n) has a primitive prime factor for all n, except n=0, 1, 2, 6 and 12. This sequence lists such indices n that A000045(n) has exactly one primitive prime factor (equal A001578(n)). Sister sequence A152013 provides indices of Fibonacci numbers with at least 2 prime factors. The current sequence A152012 and its sister sequence A152013 along with the finite set {0,1,2,6,12} form a partition of the natural numbers.
Numbers k such that A086597(k) = 1.
For prime p, all prime factors of Fibonacci(p) are primitive. Hence, the only primes in this sequence are the prime numbers in A001605, which gives the indices of prime Fibonacci numbers.

T. D. Noe, Table of n, a(n) for n=1..392

Cf. A000045, A001578, A152013.

primitivePrimeFactors[n_] := Cases[FactorInteger[Fibonacci[n]][[All, 1]], p_ /; And @@ (GCD[p, #] == 1 & /@ Array[Fibonacci, n-1])]; Reap[For[n=3, n <= 200, n++, If[Length[primitivePrimeFactors[n]] == 1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2014 *)

isok(pf, vp) = sum(i=1, #pf, vecsearch(vp, pf[i]) == 0) == 1;
lista(nn) = {vp = []; for (n=3, nn, pf = factor(fibonacci(n))[,1]; if (isok(pf, vp), print1(n, ", ")); vp = vecsort(concat(vp, pf),, 8););} \\ Michel Marcus, Nov 29 2014

A178763 Product of primitive prime factors of Fibonacci(n).

Original entry on oeis.org

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 4181, 41, 421, 199, 28657, 23, 3001, 521, 5777, 281, 514229, 31, 1346269, 2207, 19801, 3571, 141961, 107, 24157817, 9349, 135721, 2161, 165580141, 211, 433494437, 13201, 109441, 64079

Offset: 1

T. D. Noe, Jun 10 2010

nonn

Same as A001578 for the first 18 terms.

Let b(n) be the greatest divisor of Fibonacci(n) that is coprime to Fibonacci(m) for all positive integers m < n, then a(n) = b(n) for all n, provided that no Wall-Sun-Sun prime exists. Otherwise, if p is a Wall-Sun-Sun prime and A001177(p) = k (then A001177(p^2) = k), then p^2 divides b(k), but by definition a(k) is squarefree. - Jianing Song, Jul 02 2019

T. D. Noe, Table of n, a(n) for n = 1..1000
Dov Jarden, Recurring Sequences, Riveon Lematematika, Jerusalem, 1966. [Annotated scanned copy] See pp. 60-64 for a table of the first 385 terms.
Florian Luca, Carl Pomerance, and Stephen Wagner, Fibonacci Integers (preprint)

Cf. A061446, A086597, A152012 (Indices of prime terms).

a(n)=my(d=divisors(n), f=fibonacci(n), t); t=lcm(apply(fibonacci,d[1..#d-1])); while((t=gcd(t,f))>1, f/=t); f \\ Charles R Greathouse IV, Nov 30 2016

a(n) = A061446(n) / A178764(n).

a(n) = A061446(n) / gcd(A061446(n), n) if n != 5, 6, provided that no Wall-Sun-Sun prime exists. - Jianing Song, Jul 02 2019

A233281 Numbers n such that the least Fibonacci number F_k which is a multiple of n has a prime index, i.e., k is in A000040.

Original entry on oeis.org

2, 5, 13, 37, 73, 89, 113, 149, 157, 193, 233, 269, 277, 313, 353, 389, 397, 457, 557, 613, 673, 677, 733, 757, 877, 953, 977, 997, 1069, 1093, 1153, 1213, 1237, 1453, 1597, 1657, 1753, 1873, 1877, 1933, 1949, 1993, 2017, 2137, 2221, 2237, 2309, 2333, 2417, 2473

Offset: 1

Antti Karttunen, Dec 13 2013

nonn

Numbers n such that A001177(n) is prime.
Each natural number n belongs to this sequence if the smallest Fibonacci number which it divides is a term of A030426. - Jon E. Schoenfield, Feb 28 2014
A092395 gives all the primes in this sequence (cf. Wikipedia-link), and the first composite occurs as the 69th term, where a(69)=4181 while A092395(69)=4273. After 4181 (= 37*113 = F_19), the next term missing from A092395 is a(148)=10877 (= 73*149. A001177(10877) = 37, F_37 = 24157817 = 2221*10877). Both of these numbers (4181 and 10877) occur in various lists of Fibonacci-related pseudoprimes. Sequence A238082 gives all composites occurring in this sequence.
If n is in this sequence then all divisors d > 1 of n are in this sequence. - Charles R Greathouse IV, Feb 04 2014
Composite members begin 4181, 10877, 75077, 162133, 330929, .... - Charles R Greathouse IV, Mar 07 2014

Antti Karttunen and Charles R Greathouse IV, Table of n, a(n) for n = 1..2000 (first 157 terms from Karttunen)
Wikipedia, Fibonacci prime, section: Divisibility of Fibonacci numbers

Disjoint union of A092395 and A238082. The first 68 terms are identical with A092395, after which follows the first case of the latter sequence, with a(69) = A238082(1) = 4181.

Cf. A000045, A001177, A001602, A030426, A051694, A060442, A086597, A233282.

a233281 n = a233281_list !! (n-1)
a233281_list = filter ((== 1) . a010051 . a001177) [1..]
-- Reinhard Zumkeller, Apr 04 2014

is(n)=my(k); while(fibonacci(k++)%n, ); isprime(k) \\ Charles R Greathouse IV, Feb 04 2014

entry(p)=my(k=1);while(fibonacci(k++)%p,);k;
is(n)={
    if(n%2==0,return(n==2));
    if(n<13, return(n==5));
    my(f=factor(n),p,F);
    if(f[1,2]>1 && f[1,1]<1e14,return(0));
    p=entry(f[1,1]);
    F=fibonacci(p);
    if(f[1,2]>1 && F%f[1,1]^f[1,2],return(0));
    if(!isprime(p), return(0));
    for(i=2,#f~,
        if(F%f[i,1]^f[i,2],return(0))
    );
    1
}; \\ Charles R Greathouse IV, Feb 04 2014

A010051(A001177(a(n))) = 1. - Reinhard Zumkeller, Apr 04 2014

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

A174323 Numbers n such that omega(Fibonacci(n)) is a square.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 11, 13, 17, 20, 23, 24, 27, 28, 29, 32, 43, 47, 52, 55, 74, 77, 80, 83, 84, 85, 87, 88, 91, 93, 96, 97, 100, 108, 115, 123, 131, 132, 137, 138, 143, 146, 149, 156, 157, 161, 163, 178, 184, 187, 189, 196, 197, 209, 211, 214, 215, 221, 222, 223, 232

Offset: 1

Michel Lagneau, Mar 15 2010

nonn

Numbers n such that omega(A000045(n)) is a square, where omega(p) is the number of distinct prime factors of p (A001221). Remark: for the larger Fibonacci numbers F(n) (n > 300), the Maple program (below) is very slow. So we use a two-step process: factoring F(n) with the elliptic curve method, and then calculate the distinct prime factors.

			omega(Fibonacci(1)) = omega(Fibonacci(2)) = omega(1) = 0,
omega(Fibonacci(3)) = omega(2) = 1,
omega(Fibonacci(20)) = omega(6765) = 4,
omega(Fibonacci(80)) = omega(23416728348467685) = 9.

Majorie Bicknell and Verner E Hoggatt, Fibonacci's Problem Book, Fibonacci Association, San Jose, Calif., 1974.
Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.

Amiram Eldar, Table of n, a(n) for n = 1..258 (terms 1..200 from Robert Israel, derived from b-file for A022307)
Blair Kelly, Fibonacci and Lucas Factorizations
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
Eric Weisstein's World of Mathematics, Fibonacci Number
Wikipedia, Fibonacci number

Cf. A038575 (number of prime factors of n-th Fibonacci number, with multiplicity).
Cf. A000045, A000213, A000288, A000322, A000383, A001221, A060455, A030186, A039834, A020695, A020701, A071679.
Cf. A022307 (number of distinct prime factors of n-th Fibonacci number), A086597 (number of primitive prime factors).

[k:k in [1..240]| IsSquare(#PrimeDivisors(Fibonacci(k)))]; // Marius A. Burtea, Oct 15 2019

with(numtheory):u0:=0:u1:=1:for p from 2 to 400 do :s:=u0+u1:u0:=u1:u1:=s: s1:=nops( ifactors(s)[2]): w1:=sqrt(s1):w2:=floor(w1):if w1=w2 then print (p): else fi:od:
# alternative:
P[1]:= {}: count:= 1: res:= 1:
for i from 2 to 300 do
  pn:= map(t -> i/t, numtheory:-factorset(i));
  Cprimes:= `union`(seq(P[t],t=pn));
  f:= combinat:-fibonacci(i);
  for p in Cprimes do f:= f/p^padic:-ordp(f,p) od;
  P[i]:= Cprimes union numtheory:-factorset(f);
  if issqr(nops(P[i])) then
     count:= count+1;
     res:= res, i;
  fi;
od:
res; # Robert Israel, Oct 13 2016

Select[Range[200], IntegerQ[Sqrt[PrimeNu[Fibonacci[#]]]] &] (* G. C. Greubel, May 16 2017 *)

is(n)=issquare(omega(fibonacci(n))) \\ Charles R Greathouse IV, Oct 13 2016

cp's OEIS Frontend

A001605 Indices of prime Fibonacci numbers.

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A001602 Fibonacci entry points: a(n) = smallest m > 0 such that the n-th prime divides Fibonacci(m).

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A022307 Number of distinct prime factors of n-th Fibonacci number.

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A038575 Number of prime factors of n-th Fibonacci number, counted with multiplicity.

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A001578 Smallest primitive prime factor of Fibonacci number F(n), or 1 if F(n) has no primitive prime factor.

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A152012 Indices of Fibonacci numbers having exactly one primitive prime factor.

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