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

1,7

Comments

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

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

Formula

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

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

Sequence in context: A300795 A033068 A234368 * A278706 A005468 A360464

Adjacent sequences: A052008 A052009 A052010 * A052012 A052013 A052014

Keyword

nonn,nice

Author

Patrick De Geest, Nov 15 1999

Status

approved