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A093062
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a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).
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3
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-1, 0, 2, 8, 78, 214, 1556, 4108, 28518, 513972, 1345808, 24156990, 165578670, 433491846, 2971210580, 53316283380, 956722012572, 2504730758802, 44945570173074, 308061521102198, 806515532933562, 14472334024479534, 99194853094422264, 1779979416004150202
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OFFSET
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1,3
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COMMENTS
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Composition of prime( ) and Fibonacci( ) is not commutative. Does a prime p ever divide Fibonacci(prime(p)) - prime(Fibonacci(p))?
Note that a(3) = 2 is the only prime element of the sequence. This is because after 2, all primes are odd; and the Fibonacci number F(n) is even only for n = 3k for some integer k [which relates to the fact that A082115 Fibonacci sequence (mod 3) is periodic with Pisano period 8]. Hence after a(1) = -1, Fibonacci(prime(n)) - prime(Fibonacci(n)) is always the difference of two odd numbers, hence is even. - Jonathan Vos Post, Jan 23 2006
Is a(i) ever divisible by i? Answer: yes. The quotient is an integer for i = 4, 28 and 30 through 63. - Dennis S. Kluk (mathemagician(AT)ameritech.net), Aug 16 2006
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LINKS
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FORMULA
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a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).
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EXAMPLE
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a(11) = Fibonacci(prime(11)) - prime(Fibonacci(11)) = 1345808.
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MATHEMATICA
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For[i=1, i<61, i++, Print[i, " ", Fibonacci[Prime[i]]-Prime[Fibonacci[i]]]]
Table[Fibonacci[Prime[n]]-Prime[Fibonacci[n]], {n, 30}] (* Harvey P. Dale, Jul 02 2018 *)
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PROG
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(PARI) { default(primelimit, 4294965247); for(n=1, 41, a=fibonacci(prime(n)) - prime(fibonacci(n)); write("b093062.txt", n, " ", a); ); } \\ Harry J. Smith, Jun 20 2009
(Magma) [Fibonacci(NthPrime(n)) - NthPrime(Fibonacci(n)): n in [1..30]]; // Vincenzo Librandi, Apr 10 2020
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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Dennis S. Kluk (mathemagician(AT)ameritech.net), May 08 2004
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STATUS
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approved
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