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A052011
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Number of primes between successive Fibonacci numbers.
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4
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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
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OFFSET
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1,7
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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Between Fib(9)=34 and Fib(10)=55 we find the following primes: 37, 41, 43, 47 and 53 hence a(9)=5.
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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