A080345 a(n) is the number of prime factors in Fibonacci(prime(n)); that is, in the Fibonacci number whose index is the n-th prime.

0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 4, 2, 3, 2, 2, 2, 2, 1, 1, 3, 4, 2, 4, 4, 2, 2, 3, 3, 2, 2, 4, 2, 4, 4, 2, 5, 3, 4, 3, 2, 3, 3, 4, 2, 2, 3, 4, 2, 4, 4, 4, 3, 2, 3, 5, 4, 2, 1, 7, 5, 4, 3, 3, 2, 2, 4, 3, 4, 1, 1, 5, 5, 1, 3, 5, 3, 2, 3, 4, 3, 4, 6, 1, 3, 4, 3

Offset: 1

Author

T. D. Noe, Feb 16 2003

Keywords

• hard
• nonn

Comments

In all known examples, Fibonacci(prime(n)) is squarefree, in which case a(n) is well-defined, i.e., the number of distinct prime factors equals the total number of prime factors. But if for some n, Fibonacci(prime(n)) has a repeated prime factor, then a(n) is not well-defined. - Jonathan Sondow, Oct 22 2015

Examples

			a(12) = 3 because the 12th prime is 37 and Fibonacci(37) = 24157817 = 73 * 149 * 2221 has 3 prime factors. - clarified by _Jonathan Sondow_, Oct 21 2015
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..222 (terms 1..168 from T. D. Noe)
• Blair Kelly, Fibonacci and Lucas Factorizations

Crossrefs

Cf. A001605, A022307.

Programs

Mathematica

Table[Length[FactorInteger[Fibonacci[Prime[n]]]], {n, 60}]
PrimeNu[Fibonacci[Prime[Range[100]]]] (* Harvey P. Dale, Mar 13 2016 *)

PARI

a(n) = omega(fibonacci(prime(n))); \\ Michel Marcus, Oct 22 2015

Formula

a(n) = A001221(A000045(A000040(n))). - Michel Marcus, Oct 22 2015