A093062 a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).

-1, 0, 2, 8, 78, 214, 1556, 4108, 28518, 513972, 1345808, 24156990, 165578670, 433491846, 2971210580, 53316283380, 956722012572, 2504730758802, 44945570173074, 308061521102198, 806515532933562, 14472334024479534, 99194853094422264, 1779979416004150202

Offset: 1

Author

Dennis S. Kluk (mathemagician(AT)ameritech.net), May 08 2004

Keywords

• easy
• sign

Comments

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

Examples

			a(11) = Fibonacci(prime(11)) - prime(Fibonacci(11)) = 1345808.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..84 (terms 1..41 from Harry J. Smith)

Crossrefs

Cf. A000040, A000045, A030427.

Cf. A082115.

Programs

Magma

[Fibonacci(NthPrime(n)) - NthPrime(Fibonacci(n)): n in [1..30]]; // Vincenzo Librandi, Apr 10 2020

Mathematica

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 *)

PARI

a(n) = { fibonacci(prime(n)) - prime(fibonacci(n)) } \\ Harry J. Smith, Jun 20 2009

Formula

a(n) = Fibonacci(prime(n)) - prime(Fibonacci(n)).