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A052011 Number of primes between successive Fibonacci numbers. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

COMMENTS

The formula given and the sequence itself must use the same convention on "between" and what to do if one or both fibonacci numbers are themselves prime. - Jonathan Vos Post, Mar 08 2010

With the given sequence data, we see that neither endpoint is included, so we count primes p in the open interval F(n)<p<F(n+1) only. - Jeppe Stig Nielsen, Jun 06 2015

LINKS

Table of n, a(n) for n=1..43.

FORMULA

a(n) = PrimePi(F(n+1)-1) - PrimePi(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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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 *)

PROG

(Haskell)

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

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

CROSSREFS

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

Sequence in context: A300795 A033068 A234368 * A278706 A005468 A271063

Adjacent sequences:  A052008 A052009 A052010 * A052012 A052013 A052014

KEYWORD

nonn,nice

AUTHOR

Patrick De Geest, Nov 15 1999

STATUS

approved

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Last modified May 12 07:28 EDT 2021. Contains 343821 sequences. (Running on oeis4.)