A001605 -id:A001605 - cp's OEIS frontend

A080327 Numbers k for which Lucas(k) and Fibonacci(k) are both prime.

4, 5, 7, 11, 13, 17, 47, 148091

Offset: 1

T. D. Noe, Feb 15 2003

The intersection of A001605 and A001606. Fibonacci(148091) and Lucas(148091) are probable primes.
Corresponding Fibonacci-Lucas prime twins are listed in A121533. Corresponding Lucas-Fibonacci prime twins are listed in A121534. Fibonacci(148091) and Lucas(148091) are probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits. - Alexander Adamchuk, Aug 05 2006
Heuristically, this sequence is finite. It is quite probable, but presently unprovable, that it is now complete. - David Broadhurst, Jun 25 2008
Western Number Theory problem 007:13 by Gary Walsh asks to prove that a(8) = 148091 is in this sequence. - Charles R Greathouse IV, May 21 2014

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 246.

David Broadhurst, Prime Curio: 10660...49391 (61899-digits)
Gerry Myerson, Western Number Theory Problems, 17 & 19 Dec 2007.

Cf. A001605, A001606, A121533, A121534, A000045, A005478, A000032, A000204, A005479, A080327.

Select[Range[0, 100], PrimeQ[Fibonacci[#]] && PrimeQ[LucasL[#]] & ] (* Robert Price, May 27 2019 *)

is(n)=isprime(n) && ispseudoprime(fibonacci(n)) && ispseudoprime(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, May 21 2014

A121534 Lucas-Fibonacci prime twins: Prime Lucas numbers Lucas(k) such that Fibonacci numbers Fibonacci(k) are also prime.

Original entry on oeis.org

7, 11, 29, 199, 521, 3571, 6643838879

Offset: 1

Alexander Adamchuk, Aug 05 2006

nonn

Indices for Lucas-Fibonacci prime twins are A080327(n). Corresponding Fibonacci-Lucas prime twins are A121533(n). Probable primes Fibonacci(148091) and Lucas(148091) are the next probable Fibonacci-Lucas and Lucas-Fibonacci prime twins. They have 30949 and 30950 digits.

			a(1) = 7 because Lucas(4) = 7 is prime and Fibonacci(4) = 3 is prime too.

Cf. A121533, A000045, A005478, A001605, A001606, A000032, A000204, A005479, A080327.

Do[f=Fibonacci[n]; l=Fibonacci[n-1]+Fibonacci[n+1]; If[PrimeQ[f]&&PrimeQ[l], Print[{f,l}]], {n,10000}]
nn=1000;Transpose[Select[Thread[{Fibonacci[Range[nn]], LucasL[ Range[nn]]}],And@@PrimeQ[#]&]][[2]] (* Harvey P. Dale, Jul 08 2011 *)

a(1) and example corrected by Harvey P. Dale, Jul 08 2011

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

A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

Original entry on oeis.org

1, 1, 2, 3, 5, 2, 13, 3, 2, 5, 89, 2, 233, 13, 2, 3, 1597, 2, 37, 3, 2, 89, 28657, 2, 5, 233, 2, 3, 514229, 2, 557, 3, 2, 1597, 5, 2, 73, 37, 2, 3, 2789, 2, 433494437, 3, 2, 139, 2971215073, 2, 13, 5, 2, 3, 953, 2, 5, 3, 2, 59, 353, 2, 4513, 557, 2, 3, 5, 2, 269, 3, 2, 5

Offset: 1

Labos Elemer, Apr 03 2001

nonn

			For n=82: F(82) = 2789*59369*370248451, so a(82)=2789.

Tyler Busby, Table of n, a(n) for n = 1..1452 (terms 1..1000 from Alois P. Heinz, derived from Kelly's data, terms 1001..1408 from Amiram Eldar)
Blair Kelly, Fibonacci and Lucas Factorizations

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A139044.

[1,1] cat [Minimum(PrimeDivisors(Fibonacci(n))): n in [3..70]]; // Vincenzo Librandi, Dec 25 2016

f[n_] := (FactorInteger@ Fibonacci@ n)[[1,1]]; Array[f, 70] (* Robert G. Wilson v, Jul 07 2007 *)

a(n) = if ((f=fibonacci(n))==1, 1, factor(f)[1,1]); \\ Michel Marcus, Nov 15 2014

a(n) = A020639(A000045(n)). - Michel Marcus, Nov 15 2014

Better definition from Omar E. Pol, Apr 25 2008

A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937).

Original entry on oeis.org

37, 73, 113, 149, 157, 193, 269, 277, 313, 353, 389, 397, 457, 557, 613, 673, 677, 733, 757, 877, 953, 977, 997, 1069, 1093, 1153, 1213, 1237, 1453, 1657, 1753, 1873, 1877, 1933, 1949, 1993, 2017, 2137, 2221, 2237, 2309, 2333, 2417, 2473, 2557, 2593, 2749, 2777, 2789, 2797, 2857, 2909, 2917, 3217, 3253, 3313, 3517, 3557, 3733, 4013, 4057, 4177, 4273, 4349, 4357, 4513, 4637, 4733, 4909, 4933

Offset: 1

Artur Jasinski, Dec 08 2007

nonn

All numbers in this sequence are congruent to 1 mod 4. - Max Alekseyev.
If Fibonacci(n) is divisible by a prime p of the form 4k+3 then n is even. To prove this statement it is enough to show that (1+sqrt(5))/(1-sqrt(5)) is never a square modulo such p (which is a straightforward exercise).
The n-th prime p is an element of this sequence iff A001602(n) is prime and A051694(n)=A000045(A001602(n))>p. - Max Alekseyev

Hans Havermann, Table of n, a(n) for n = 1..5000

Cf. A000045, A001605, A050937, A075737, A090819, A134787, A134851, A134852.

a = {}; k = {}; Do[If[ !PrimeQ[Fibonacci[Prime[n]]], s = FactorInteger[Fibonacci[Prime[n]]]; c = Length[s]; Do[AppendTo[k, s[[m]][[1]]], {m, 1, c}]], {n, 2, 60}]; Union[k]

Edited, corrected and extended by Max Alekseyev, Dec 12 2007

A153887 Smallest of five consecutive Fibonacci numbers whose sum is a prime number.

Original entry on oeis.org

0, 1, 2, 8, 89, 144, 2584, 121393, 196418, 832040, 20365011074, 4052739537881, 8944394323791464, 99194853094755497, 2880067194370816120, 14028366653498915298923761, 96151855463018422468774568

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

0+1+1+2=3=7, 1+2+3+5+8=19, 2+3+5+8=13=31, 8+13+21+34+55=131, 89+144+233+377+610=1453, 144+233+377+610+987=2351,...

			For n=3,4, the Fibonacci indices of a(3)=2, a(4)=8, are 3,6 respectively. So a(3) + A000032(7)= 31,  a(4) + A000032(10) = 131. - _Vladimir Shevelev_, Apr 24 2014

Cf. A000045, A001906, A000071, A001605, A153862, A153863, A153865, A153866, A153867.

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,a]];a=b;b=c;c=d;d=e,{n,4,6!}];lst
Select[Partition[Fibonacci[Range[1000]],5,1],PrimeQ[Total[#]]&][[All,1]] (* Harvey P. Dale, Dec 01 2016 *)

For n>=3, a(n) + A000032(m+4) is prime, where m is the Fibonacci index of a(n) (see example). This follows from the identity F(m+1) + F(m+2) + F(m+3) + F(m+4) = A000032(m+4), m>=0. - Vladimir Shevelev, Apr 24 2014

A153888 Second-to-smallest of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

Original entry on oeis.org

1, 2, 3, 13, 144, 233, 4181, 196418, 317811, 1346269, 32951280099, 6557470319842, 14472334024676221, 160500643816367088, 4660046610375530309, 22698374052006863956975682, 155576970220531065681649693

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

nonn

0+1+1+2=3=7, 1+2+3+5+8=19, 2+3+5+8=13=31, 8+13+21+34+55=131, 89+144+233+377+610=1453, 144+233+377+610+987=2351,...

Harvey P. Dale, Table of n, a(n) for n = 1..29

Cf. A000045, A001906, A000071, A001605, A153862, A153863, A153865, A153866, A153867, A153887

a=0;b=1;c=1;d=2;lst={};Do[e=Fibonacci[n];p=a+b+c+d+e;If[PrimeQ[p],AppendTo[lst,b]];a=b;b=c;c=d;d=e,{n,4,6!}];lst
Select[Partition[Fibonacci[Range[0,150]],5,1],PrimeQ[Total[#]]&][[All,2]] (* Harvey P. Dale, Dec 11 2018 *)

A281087 Numbers k such that Fibonacci(k) and Fibonacci(k+2) are both prime.

Original entry on oeis.org

3, 5, 11, 431, 569

Offset: 1

Bobby Jacobs, Jan 14 2017

Smaller primes of the Fibonacci prime pairs in A073340.

See the comment to A073340 - Harvey P. Dale, Jan 30 2025

			11 is in the sequence because Fibonacci(11) = 89 and Fibonacci(13) = 233 are both prime.

J. B. Cosgrave and K. Dilcher, Pairs of reciprocal quadratic congruences involving primes, Fig. Quart. 51 (2) (2013) 98-111

First differs from A101315 at a(5).

Cf. A000045, A001359, A001605, A073340, A101315, A279795.

Select[Range[600],PrimeQ[Fibonacci[#]] && PrimeQ[Fibonacci[#+2]] &] (* Stefano Spezia, Nov 15 2024 *)
SequencePosition[Table[If[PrimeQ[Fibonacci[n]],1,0],{n,600}],{1,,1}][[;;,1]] (* _Harvey P. Dale, Jan 30 2025 *)

a(n) = A279795(n) - 2.

a(n) = A073340(2n-1).

A052011 Number of primes between successive Fibonacci numbers exclusive.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 16, 23, 37, 55, 84, 125, 198, 297, 458, 704, 1087, 1673, 2602, 4029, 6263, 9738, 15186, 23704, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419527, 13298630

Offset: 1

Patrick De Geest, Nov 15 1999

With the given sequence data, we see that neither endpoint is included, so we count primes p in the open interval F(n)Jeppe Stig Nielsen, Jun 06 2015

			Between Fib(9)=34 and Fib(10)=55 we find the following primes: 37, 41, 43, 47 and 53 hence a(9)=5.

Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated from the data at A054782 and A001605)

Cf. A000040, A001605, A005478 (endpoint primes), A010051, A052012, A054782.

a052011 n = a052011_list !! (n-1)
a052011_list = c 0 0 $ drop 2 a000045_list where
  c x y fs'@(f:fs) | x < f     = c (x+1) (y + a010051 x) fs'
                   | otherwise = y : c (x+1) 0 fs
-- Reinhard Zumkeller, Dec 18 2011

for n from 1 to 43 do T[n]:= numtheory:-pi(combinat:-fibonacci(n)) od:
seq(T[n]-T[n-1]-`if`(isprime(combinat:-fibonacci(n)),1,0), n=2..43); # Robert Israel, Jun 08 2015

lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n]+1,Fibonacci[n+1]-1}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
pbf[n_]:=Module[{fib1=If[PrimeQ[Fibonacci[n+1]],PrimePi[Fibonacci[n+1]-1], PrimePi[ Fibonacci[n+1]]], fib0=If[PrimeQ[Fibonacci[n]], PrimePi[ Fibonacci[n]+1],PrimePi[Fibonacci[n]]]},Max[0,fib1-fib0]]; Array[pbf,50] (* Harvey P. Dale, Mar 01 2012 *)

a(n)=my(s); forprime(p=fibonacci(n)+1,fibonacci(n+1)-1,s++); s \\ Charles R Greathouse IV, Jun 08 2015

a(n) = pi(F(n+1)-1) - pi(F(n)) = A000720(A000045(n+1)-1) - A000720(A000045(n)). - Jonathan Vos Post, Mar 08 2010; corrected by Jeppe Stig Nielsen, Jun 06 2015
a(n) ~ phi^(n-1)/(n*sqrt(5)*log(phi)), where phi = (1+sqrt(5))/2 is the golden ratio. - Charles R Greathouse IV, Jun 08 2015
a(n) = A054782(n+1) - A054782(n) - [n+1 in A001605], where [] denotes the Iverson bracket. - Amiram Eldar, Jun 10 2024

A076777 Number of primes between successive Fibonacci numbers inclusive.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 2, 2, 3, 5, 8, 10, 17, 23, 37, 55, 85, 125, 198, 297, 458, 704, 1088, 1673, 2602, 4029, 6263, 9738, 15187, 23704, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419528, 13298630, 21014892, 33227992

Offset: 0

Reinhard Zumkeller, Nov 14 2002

nonn

a(n) = #{p prime | A000045(n)A000045(n+1)}.

			a(10) = 8, as there are 8 primes greater than A000045(10) = 55 and not greater than A000045(10+1) = 89: 59, 61, 67, 71, 73, 79, 83 and 89.

Amiram Eldar, Table of n, a(n) for n = 0..122 (calculated using the b-file at A054782)

Cf. A000045, A000720, A054782, A082602.

Cf. A001605, A005478.

with(combinat): with(numtheory): seq(pi(fibonacci(n+1))-pi(fibonacci(n)),n=0..35); # Emeric Deutsch

Table[PrimePi[Fibonacci[k+1]]-PrimePi[Fibonacci[k]],{k,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 30 2010 *)

A076777(n) = primepi(fibonacci(n+1))-primepi(fibonacci(n))
A076777(n) = sum(i=fibonacci(n)+1,fibonacci(n+1),isprime(i)) \\ Michael B. Porter, Nov 24 2009

a(n) = A000720(A000045(n+1)) - A000720(A000045(n)).

More terms from Emeric Deutsch, Mar 02 2005

More terms from Amiram Eldar, Oct 07 2021

cp's OEIS Frontend

A080327 Numbers k for which Lucas(k) and Fibonacci(k) are both prime.

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A121534 Lucas-Fibonacci prime twins: Prime Lucas numbers Lucas(k) such that Fibonacci numbers Fibonacci(k) are also prime.

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

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A060383 a(1) = a(2) = 1; for n >2, a(n) = smallest prime factor of n-th Fibonacci number.

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A135952 Prime factors of composite Fibonacci numbers with prime indices (cf. A050937).

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A153887 Smallest of five consecutive Fibonacci numbers whose sum is a prime number.

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A153888 Second-to-smallest of five consecutive Fibonacci numbers such that sum of five consecutive Fibonacci numbers is prime number.

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A281087 Numbers k such that Fibonacci(k) and Fibonacci(k+2) are both prime.

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