A030426 -id:A030426 - cp's OEIS frontend

A325626 a(n) is the smallest prime factor in A030426(n).

2, 5, 13, 89, 233, 1597, 37, 28657, 514229, 557, 73, 2789, 433494437, 2971215073, 953, 353, 4513, 269, 6673, 9375829, 157, 99194853094755497, 1069, 193, 743519377, 519121, 1247833, 827728777, 677, 27941, 1066340417491710595814572169, 19134702400093278081449423917, 277, 110557

Offset: 2

Vincenzo Librandi, May 11 2019

nonn

Chai Wah Wu, Table of n, a(n) for n = 2..230

Cf. A000045, A005478, A030426.

[Minimum(PrimeDivisors(Fibonacci(NthPrime(n)))): n in [2..35]];

f[n_]:=(FactorInteger@Fibonacci@Prime[n])[[1, 1]]; Array[f, 35, 2]

Offset corrected by Chai Wah Wu, Jul 17 2019

A325627 a(n) is the largest prime factor in A030426(n).

Original entry on oeis.org

2, 5, 13, 89, 233, 1597, 113, 28657, 514229, 2417, 2221, 59369, 433494437, 2971215073, 55945741, 2710260697, 555003497, 1429913, 46165371073, 86020717, 92180471494753, 99194853094755497, 1665088321800481, 361040209, 770857978613, 512119709, 8242065050061761

Offset: 1

Vincenzo Librandi, May 13 2019

nonn

Amiram Eldar, Table of n, a(n) for n = 1..222
C. L. Stewart, On Divisors of Fermat, Fibonacci, Lucas, and Lehmer Numbers, Proceedings of the London Mathematical Society, Vol. s3-35, No. 3 (1977), pp. 425-447. See p. 430.

Cf. A000045, A005478, A006530, A030426, A060385, A325626.

[Maximum(PrimeDivisors(Fibonacci(NthPrime(n)))): n in [2..35]];

Table[FactorInteger[Fibonacci [Prime[n]]][[-1, 1]], {n, 2, 30}]

From Amiram Eldar, Oct 25 2024: (Start)
a(n) = A006530(A030426(n)).
a(n) = A060385(prime(n+1)).
a(n) > c * prime(n) * log(prime(n)), where c is an effectively computable positive constant (Stewart, 1977). (End)

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

A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.

Original entry on oeis.org

0, 0, 1, 2, 4, 5, 7, 13, 12, 12, 15, 89, 20, 233, 33, 25, 32, 1597, 33, 4181, 40, 53, 189, 28657, 52, 50, 479, 54, 80, 514229, 65, 1346269, 80, 289, 3211, 100, 84, 24157817, 8381, 725, 100, 165580141, 127, 433494437, 400, 105, 57337, 2971215073, 128, 182, 125, 4825, 984, 53316291173, 135, 500, 188, 12581, 1028487, 956722026041, 160

Offset: 0

Antti Karttunen, Oct 28 2019

nonn

Antti Karttunen, Table of n, a(n) for n = 0..1001

Cf. A000040, A000045, A007895, A030426, A113175, A328847, A374201.
Cf. A374046 (indices of even terms), A374047 (of odd terms), A374122 (of multiples of 3), A374202 (2-adic valuation), A374203 (3-adic valuation), A374205 (5-adic valuation), A374125 [a(n) mod 360].
Cf. A374106 [gcd(a(n), A113177(n))], A374035 [gcd(a(n), A328846(n))], A374116 [gcd(a(n), A328768(n))].
For variants of the same formula, see A003415, A258851, A328768, A328769, A328846, A371192.

A328845[n_] := If[n <= 1, 0, n*Total[MapApply[#2*Fibonacci[#]/# &, FactorInteger[n]]]];
Array[A328845, 100, 0] (* Paolo Xausa, Dec 16 2024 *)

A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));

a(n) = n * Sum e_j * A000045(p_j)/p_j for n = Product p_j^e_j.
a(A000040(n)) = A030426(n).
A007895(a(n)) = A328847(n).

A075737 Prime Fibonacci numbers with prime indices.

Original entry on oeis.org

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

Offset: 1

Jani Melik, Oct 07 2002

nonn

Same as A005478 except that F(4) = 3 has been omitted.

Sequence of primes in A001519. [James R. Buddenhagen, May 20 2010]

			5 is a prime and fibonacci(5)=5 is also a prime, 7 is a prime and fibonacci(7)=13 is also a prime, but 2 is a prime and fibonacci(2)=1 is not a prime.

T. D. Noe, Table of n, a(n) for n = 1..20

Subsequence of A030426.

Cf. A000045, A005478, A083668, A001519.

with(combinat, fibonacci): fib_supM_pra := proc(n); if (isprime(n)='true') then if (isprime(fibonacci(n))='true') then RETURN(fibonacci(n)); fi; fi; end: seq(fib_supM_pra(i), i=1..500);

Fibonacci[ Prime[ Select[ Range[50], PrimeQ[ Fibonacci[ Prime[ # ]]] & ]]]
Module[{nn=500,fibs},fibs=Fibonacci[Range[nn]];Select[Pick[fibs,Table[ If[ PrimeQ[n],1,0],{n,nn}],1],PrimeQ]] (* Harvey P. Dale, Sep 13 2018 *)

forprime(p=2,1e3,if(isprime(t=fibonacci(p)), print1(t", "))) \\ Charles R Greathouse IV, Feb 03 2014

A061305 Squarefree Fibonacci numbers.

Original entry on oeis.org

1, 1, 2, 3, 5, 13, 21, 34, 55, 89, 233, 377, 610, 987, 1597, 4181, 6765, 10946, 17711, 28657, 121393, 196418, 317811, 514229, 1346269, 2178309, 3524578, 5702887, 9227465, 24157817, 39088169, 63245986, 102334155, 165580141, 433494437, 701408733, 1134903170, 1836311903

Offset: 1

Amarnath Murthy, Apr 26 2001

Union of A074691 and A075735. - R. J. Mathar, Feb 06 2010
About 60% of entries are of the form 4k+1; 20% are of the form 4k+2; 20% are of the form 4k+3. Obviously no term is divisible by 4. - Carmine Suriano, Feb 27 2014
Contains A030426 as a subsequence unless there exist Wall-Sun-Sun primes. - Max Alekseyev, Jan 04 2018

			55 = 5 * 11 is a squarefree Fibonacci number.

Amiram Eldar, Table of n, a(n) for n = 1..1080 (terms 1..200 from Zak Seidov)

Intersection of A000045 and A005117.

a={}; Do[f=Fibonacci[n]; If[SquareFreeQ[f], AppendTo[a, f]], {n, 1, 50}]; a (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
Select[Fibonacci[Range[50]],SquareFreeQ] (* Harvey P. Dale, Aug 26 2021 *)

{ n=0; g=0; f=1; for (i=1, 500, if (issquarefree(g), write("b061305.txt", n++, " ", g)); if (n==200, break); s=f; f+=g; g=s ) } \\ Harry J. Smith, Jul 21 2009

Set difference of A000045 and A061899. a(n) = A000045(A037918(n)). - R. J. Mathar, Feb 16 2010

More terms from Asher Auel, May 14 2001

Mathematica updated by Jean-François Alcover, Jul 04 2013

A119984 Numbers k such that Fibonacci(prime(k)) is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 14, 15, 23, 32, 33, 72, 83, 84, 87, 97, 104, 105, 429, 637, 710, 1152, 1194, 1692, 2814, 3316, 3824, 3971, 5206, 8002, 10016, 12161, 13681, 18069, 33653, 36467, 48355, 48629, 49455, 73574, 82128, 99005, 123685, 135276, 146779, 210404, 233207, 239581

Offset: 1

Alexander Adamchuk, Aug 04 2006

nonn

All prime Fibonacci numbers have prime indices, except prime F(4) = 3; a(n) is such that Fibonacci(prime(a(n))) is prime. - Robert G. Wilson v, Aug 05 2006

Amiram Eldar, Table of n, a(n) for n = 1..55

Cf. A000045, A000720, A001605, A005478, A030426, A075737, A083668.

Select[ Range@3000, PrimeQ@ Fibonacci@ Prime@ # &] (* Robert G. Wilson v, Aug 05 2006 *)

a(n) = pi(A001605(n+1)). This holds for all n including n=1, since pi(4) = pi(3) = 2. - Jens Kruse Andersen, Jul 24 2014

a(21)-a(27) from Robert G. Wilson v, Aug 05 2006
More terms (from A001605) from T. D. Noe, Aug 18 2006
a(42)-a(48) (from A001605, found by Henri Lifchitz) from Jens Kruse Andersen, Jul 24 2014
a(49)-a(50) (from A001605, found by Henri Lifchitz) from Amiram Eldar, Sep 01 2019

A037917 Numbers n such that the Fibonacci number F(n) is divisible by a square.

Original entry on oeis.org

6, 12, 18, 24, 25, 30, 36, 42, 48, 50, 54, 56, 60, 66, 72, 75, 78, 84, 90, 91, 96, 100, 102, 108, 110, 112, 114, 120, 125, 126, 132, 138, 144, 150, 153, 156, 162, 168, 174, 175, 180, 182, 186, 192, 198, 200, 204, 210, 216, 220, 222, 224, 225, 228, 234, 240

Offset: 1

Marc LeBrun

nonn

Is a(n) asymptotic to C*n with 4 < C < 4.5 ? - Benoit Cloitre, Sep 04 2002
Numbers are a superset of the multiples of 6 (A008588), because 8 divides Fibonacci(6m) = A134492(m). Sequence apparently also contains the multiples of 25. Are all a(n) composite? Members not divisible by 6 or 25 are 56, 91, 110, 112, 153, 182, 220, 224, 273, 280, ... - Ralf Stephan, Jan 26 2014
These numbers are the positive multiples of A065069. - Charles R Greathouse IV, Feb 02 2014
To address Cloitre's question, if such C exists it must be less than 4.3 using the known terms of A065069. - Charles R Greathouse IV, Feb 04 2014

Amiram Eldar, Table of n, a(n) for n = 1..328 (terms 1..235 from T. D. Noe, based on Kelly's table)
Blair Kelly, Fibonacci Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number.

Cf. A000045, A037918, A001177, A030426.

Select[Range[100], MoebiusMu[Fibonacci[#]] == 0 &] (* Alonso del Arte, Jan 26 2014 *)

is(n)=!issquarefree(fibonacci(n)) \\ Charles R Greathouse IV, Feb 02 2014

More terms from Eric W. Weisstein

A374106 a(n) = gcd(A113177(n), A328845(n)), where A113177 is fully additive with a(p) = Fibonacci(p) and A328845 is the first Fibonacci-based variant of the arithmetic derivative.

Original entry on oeis.org

0, 1, 2, 2, 5, 1, 13, 3, 4, 3, 89, 4, 233, 1, 1, 4, 1597, 1, 4181, 1, 1, 9, 28657, 1, 10, 1, 6, 5, 514229, 1, 1346269, 5, 1, 1, 2, 6, 24157817, 17, 5, 4, 165580141, 1, 433494437, 1, 3, 7, 2971215073, 2, 26, 1, 1, 1, 53316291173, 1, 2, 4, 1, 3, 956722026041, 1, 2504730781961, 1, 1, 6, 2, 1, 44945570212853, 3, 1, 1

Offset: 1

Antti Karttunen, Jun 28 2024

nonn

Antti Karttunen, Table of n, a(n) for n = 1..4500

Cf. A000040, A000045, A030426, A113177, A328845.

Cf. also A374035.

A113177(n) = if(n<=1, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])));
A328845(n) = if(n<=1, 0, my(f=factor(n)); n*sum(i=1, #f~, f[i, 2]*fibonacci(f[i,1])/f[i, 1]));
A374106(n) = gcd(A113177(n), A328845(n));

For all n >= 1, a(A000040(n)) = A030426(n).

A180363 a(n) = Lucas(prime(n)).

Original entry on oeis.org

3, 4, 11, 29, 199, 521, 3571, 9349, 64079, 1149851, 3010349, 54018521, 370248451, 969323029, 6643838879, 119218851371, 2139295485799, 5600748293801, 100501350283429, 688846502588399, 1803423556807921, 32361122672259149, 221806434537978679

Offset: 1

Jonathan Vos Post, Aug 31 2010

This is to A030426, Fibonacci(prime(n)), as A000032 (Lucas numbers beginning at 2) is to A000045.

			a(1) = 3 because the 1st prime is 2, and the 2nd Lucas number is A000032(2) = 3.
a(2) = 4 because the 2nd prime is 3, and the 3rd Lucas number is A000032(3) = 4.
a(3) = 11 because the 3rd prime is 5, and the 5th Lucas number is A000032(5) = 11.

Table of n, a(n) for n = 1..650
A. Aksenov, The Newman phenomenon and Lucas sequence, arXiv:1108.5352 [math.NT], 2011. [Gives factorizations of first 88 terms]
Paula Burkhardt et al., Visual properties of generalized Kloosterman sums, arXiv:1505.00018 [math.NT], 2015.

Cf. A000032, A000040, A000045, A030426.

[Lucas(NthPrime(n)): n in [1..30]]; // Vincenzo Librandi, Dec 01 2015

A180363 := proc(n) A000032(ithprime(n)) ; end proc: seq(A180363(n),n=1..30) ; # R. J. Mathar, Sep 01 2010
# second Maple program:
a:= n-> (<<1|1>, <1|0>>^ithprime(n). <<2, -1>>)[1, 1]:
seq(a(n), n=1..50);  # Alois P. Heinz, Jan 03 2022

LucasL[Prime[Range[30]]] (* Vincenzo Librandi, Dec 01 2015 *)

from sympy import lucas, prime
def a(n): return lucas(prime(n))
print([a(n) for n in range(1, 24)]) # Michael S. Branicky, Dec 30 2021

a(n) = A000032(A000040(n)) = Lucas(prime(n)).

a(n) = A032170(A000040(n)) / A064723(n-1) - 1 for n>1. - Flávio V. Fernandes, Dec 30 2021

Entries checked by R. J. Mathar, Sep 01 2010

Edited by N. J. A. Sloane, Nov 28 2011

cp's OEIS Frontend

A325626 a(n) is the smallest prime factor in A030426(n).

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A325627 a(n) is the largest prime factor in A030426(n).

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A005478 Prime Fibonacci numbers.

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A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(u*v) = a(u)*v + u*a(v), with a(0) = a(1) = 0.

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A075737 Prime Fibonacci numbers with prime indices.

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A061305 Squarefree Fibonacci numbers.

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A119984 Numbers k such that Fibonacci(prime(k)) is prime.

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A328845 The first Fibonacci based variant of arithmetic derivative: a(p) = A000045(p) for prime p, a(uv) = a(u)v + u*a(v), with a(0) = a(1) = 0.