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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
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
Amiram Eldar, Table of n, a(n) for n = 1..122 (calculated from the data at A054782 and A001605)
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
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
KEYWORD
nonn,nice
AUTHOR
Patrick De Geest, Nov 15 1999
STATUS
approved