A005478 -id:A005478 - cp's OEIS frontend

A052011 Number of primes between successive Fibonacci numbers exclusive.

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

A092579 A sieve using the Fibonacci sequence over the integers >=2. Any multiple of a Fibonacci number, F(n)*m, such that F(n)>=2 and m>=2 is excluded and what is left is included.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 97, 101, 103, 107, 109, 113, 119, 121, 127, 131, 133, 137, 139, 149, 151, 157, 161, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 203, 209, 211, 217, 223, 227, 229

Offset: 1

Christer Mauritz Blomqvist (MauritzTortoise(AT)hotmail.com), Apr 09 2004

The first number in this sequence that differs from the sequence of primes is 49. This sequence will include more and more nonprime numbers since the density of this sequence nearly linear with just a bit below one number in four included in the sequence.

The density of numbers in the sequence will approach 1/4.129112110113143678897 = The limit of the product of the terms (1-1/pf(n)) as n goes from 1 to infinity and pf(n) is the prime Fibonacci numbers (A005478).

			The number 23 is included since it is not of the form F(n)*m, F(n)>=2, m>=2. The number 21 is excluded since 21=F(4)*7=3*7.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000045, A000040, A005478.

fs[s_] := (t = Floor[s/2]; v = Range[s]; f1 = 1; f2 = 1; While[f2 < t, f = f1 + f2; f1 = f2; f2 = f; n = 2*f2; While[n <= s, v[[n]] = 0; n = n + f2]]; Select[v, #>1 &]) (* This will generate all numbers in the sequence <=s. *)

A099000 Indices k such that the k-th prime is a Fibonacci number.

Original entry on oeis.org

1, 2, 3, 6, 24, 51, 251, 3121, 42613, 23023556, 143130479, 2602986161967491

Offset: 1

Rick L. Shepherd, Nov 06 2004

From Hugo Pfoertner, Jan 06 2020: (Start)
The computation of the next two terms, corresponding to the primes F(131) = A005478(13) = 1066340417491710595814572169, and F(137) = A005478(14) = 19134702400093278081449423917, should already be within reach with current (2020) technology, e.g. with Kim Walisch's "primecount" program, which allows massive parallelization. An exact determination of the following term a(15), which corresponds to F(359), is beyond any imaginable technical possibility.
Estimates for a(13)-a(15), found by using the PARI program from A121046 in a bisection loop, with an accuracy that corresponds to the shown number of digits, are as follows:
a(13) = primepi(F(131)) ~= 1.741898800848...*10^25,
a(14) = primepi(F(137)) ~= 2.9848914766265...*10^26,
a(15) = primepi(F(359)) ~= 2.78114064956041656819790214151422895...*10^72.
(End)

Cf. A001605 (n-th Fibonacci number is prime), A005478 (Prime Fibonacci numbers).

Cf. A121046.

PrimePi[Select[Fibonacci[Range[80]], PrimeQ]]

print1("1, 2");forprime(p=5,47,if(isprime(fibonacci(p)),print1(", "primepi(fibonacci(p))))) \\ Charles R Greathouse IV, Aug 21 2011

a(n) = A000720(A005478(n)). - M. F. Hasler, Aug 21 2011

a(11) from Ryan Propper, Oct 16 2005

a(12) from Charles R Greathouse IV, Aug 21 2011

A122534 Numbers k such that Fibonacci(prime(prime(k))) is prime.

Original entry on oeis.org

1, 2, 3, 4, 9, 23, 25, 1456, 1616, 3865

Offset: 1

Alexander Adamchuk, Sep 18 2006

The corresponding primes are {2,5,89,1597,99194853094755497,...}.
Numbers k such that A093308(k) is prime.
A277575(n) = prime(a(n)) is a prime in A119984.

Cf. A000040, A000045, A119984, A083668, A093308, A075737, A001605, A006450, A005478, A030426, A277284, A277575.

a(n) = PrimePi(A277575(n)) = PrimePi(PrimePi(A277284(n))). - Bobby Jacobs, Oct 26 2016

A163853 Primes 4 less than some Fibonacci number.

Original entry on oeis.org

17, 229, 373, 983, 4177, 6761, 17707, 4052739537877, 190392490709131, 19740274219868223163, 354224848179261915071, 11825896447871834976429068423, 3807901929474025356630904134047

Offset: 1

Vladimir Joseph Stephan Orlovsky, Aug 05 2009

nonn

Primes of the form A000045(i)-4, generated by i= 8, 13, 14, 16, 19, 20, 22,...

representing prime(j) at j= 4, 7, 50, 74, 166, 574, 870, 2033,...

Harvey P. Dale, Table of n, a(n) for n = 1..23

Cf. A000045, A005478, A163851, A163852, A163854, A163855

Clear[lst,a,f,n,p]; a=4;lst={};Do[f=Fibonacci[n];If[PrimeQ[p=f-a]&&p> 1,AppendTo[lst,p]],{n,3*6!}];lst
Select[Fibonacci[Range[4,300]]-4,PrimeQ] (* Harvey P. Dale, Jul 12 2025 *)

Some indices to Fibonacci and prime sequences added by R. J. Mathar, Sep 17 2009

A178762 Prime numbers that are Fibonacci integers.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 61, 89, 107, 199, 211, 233, 281, 421, 521, 1103, 1597, 2161, 2207, 2521, 3001, 3571, 5779, 9349, 9901, 14503, 19801, 28657, 90481, 103681, 109441, 135721, 141961, 514229, 3010349, 6376021, 11128427

Offset: 1

T. D. Noe, Jun 10 2010

nonn

A Fibonacci integer is a number that can be written as the product and/or quotient of Fibonacci numbers (A000045). For example, 107 is a Fibonacci integer because Fib(36)/(Fib(18)*Fib(3)*Fib(4)^3) = 107. Observe that the prime Fibonacci numbers (A005478) are a subset of these primes. Luca, Pomerance, and Wagner conjecture that this sequence is infinite. The paper's Remark 2 and sequences A152012, A178763, and A178764 are useful in finding these primes.

Florian Luca, Carl Pomerance, and Stephan Wagner, Fibonacci Integers (preprint)

A256222 Largest Fibonacci number in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

Original entry on oeis.org

0, 1, 3, 5, 13, 13, 13, 89, 89, 89, 1597, 1597, 1597, 1597, 1597, 1597, 17711, 17711, 17711, 28657, 28657, 28657, 28657, 1346269, 1346269, 1346269, 1346269, 24157817, 24157817, 24157817, 24157817, 24157817, 24157817, 39088169, 39088169, 39088169, 39088169

Offset: 0

Michel Lagneau, Mar 19 2015

nonn

The prime Fibonacci numbers in the sequence are 3, 5, 13, 89, 1597, 28657, ...

For information about how often the numerator of these sums is a Fibonacci number, see A256220 and A256221.

			a(3) = 5 because we obtain the 5 subsets {1}, {1/2}, {1/3}, {1,1/2} and {1/2, 1/3} having 5 sums with Fibonacci numerators: 1, 1, 1, 1+1/2 = 3/2 and 1/2+1/3 = 5/6 => the greatest Fibonacci number is 5.

Hiroaki Yamanouchi, Table of n, a(n) for n = 0..50

Cf. A000045, A005478, A075226, A256220, A256221, A256223.

<<"DiscreteMath`Combinatorica`"; maxN=24; For[t={}; mx=0; i=0; n=0, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[IntegerQ[Sqrt[5*k^2+4]]||IntegerQ[Sqrt[5*k^2-4]], If[k>mx, t=s]; mx=Max[mx, k]]]; Print[mx]]

Corrected and extended by Alois P. Heinz, Mar 25 2015

a(30)-a(36) from Hiroaki Yamanouchi, Mar 30 2015

A082602 Number of primes between successive Fibonacci numbers (including possibly the Fibonacci numbers themselves).

Original entry on oeis.org

0, 1, 2, 2, 2, 2, 3, 3, 5, 8, 11, 17, 24, 37, 55, 85, 126, 198, 297, 458, 704, 1088, 1674, 2602, 4029, 6263, 9738, 15187, 23705, 36981, 57909, 90550, 142033, 222855, 349862, 549903, 865019, 1361581, 2145191, 3381318, 5334509, 8419528, 13298631, 21014892

Offset: 1

Hauke Worpel (hw1(AT)email.com), May 23 2003

nonn

			a(10) = 8 because the 10th Fibonacci number is 55, the 11th is 89 and the eight primes between them are 59, 61, 67, 71, 73, 79, 83 and 89.

Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated using the b-file at A054782)

Cf. A000045, A054782, A076777.

Cf. A001605, A005478.

[#PrimesInInterval(Fibonacci(n-1), Fibonacci(n)): n in [2..45]]; // Vincenzo Librandi, Jul 13 2017

lst={};Do[p=0;Do[If[PrimeQ[a],p++ ],{a,Fibonacci[n],Fibonacci[n+1]}];AppendTo[lst,p],{n,50}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)

{ a(n)= c=0; forprime(N=fibonacci(n),fibonacci(n+1),c=c+1); return(c); }

Corrected and extended by Rick L. Shepherd, May 26 2003

a(43)-a(44) from Vincenzo Librandi, Jul 13 2017

A105758 Indices of prime hexanacci (or Fibonacci 6-step) numbers A001592 (using offset -4).

Original entry on oeis.org

3, 36, 37, 92, 660, 6091, 8415, 11467, 13686, 38831, 49828, 97148

Offset: 1

T. D. Noe, Apr 22 2005

No other n < 30000.
This sequence uses the convention of the Noe and Post reference. Their indexing scheme differs by 4 from the indices in A001592. Sequence A249635 lists the indices of the same primes (A105759) using the indexing scheme as defined in A001592. - Robert Price, Nov 02 2014 [Edited by M. F. Hasler, Apr 22 2018]
a(13) > 3*10^5. - Robert Price, Nov 02 2014

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 n-Step Number

Cf. A105759 (prime Fibonacci 6-step numbers), A249635 (= a(n) + 4), A001592.
Cf. A000045, A000073, A000078 (and A001631), A001591, A122189 (or A066178),  A079262, A104144,  A122265, A168082, A168083 (Fibonacci, tribonacci, tetranacci numbers and other generalizations).
Cf. A005478, A092836, A104535, A105757, A105761, ... (primes in these sequence).
Cf. A001605, A303263, A303264 (and A104534 and A247027), A248757 (and A105756), ... (indices of primes in A000045, A000073, A000078, ...).

a={1, 0, 0, 0, 0, 0}; lst={}; Do[s=Plus@@a; a=RotateLeft[a]; a[[ -1]]=s; If[PrimeQ[s], AppendTo[lst, n]], {n, 30000}]; lst

a(n) = A249635(n) - 4. A105759(n) = A001592(A249635(n)) = A001592(a(n) + 4). - M. F. Hasler, Apr 22 2018

a(10)-a(12) from Robert Price, Nov 02 2014

Edited by M. F. Hasler, Apr 22 2018

cp's OEIS Frontend

A052011 Number of primes between successive Fibonacci numbers exclusive.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A076777 Number of primes between successive Fibonacci numbers inclusive.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A092579 A sieve using the Fibonacci sequence over the integers >=2. Any multiple of a Fibonacci number, F(n)*m, such that F(n)>=2 and m>=2 is excluded and what is left is included.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A099000 Indices k such that the k-th prime is a Fibonacci number.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A122534 Numbers k such that Fibonacci(prime(prime(k))) is prime.

Views

Author

Keywords

Comments

Crossrefs

Formula

A163853 Primes 4 less than some Fibonacci number.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A178762 Prime numbers that are Fibonacci integers.

Views

Author

Keywords

Comments

Links

A256222 Largest Fibonacci number in the numerator of the 2^n sums generated from the set 1, 1/2, 1/3, ..., 1/n.

Views

Author

Keywords

Comments

Examples

Links