A271327 Integers k such that the k-th prime divides the product of the first k nonzero Fibonacci numbers.

18, 24, 29, 30, 46, 47, 51, 60, 63, 71, 82, 89, 98, 100, 102, 121, 123, 127, 135, 136, 139, 145, 149, 152, 156, 157, 162, 163, 165, 169, 180, 181, 184, 185, 221, 225, 235, 251, 252, 313, 316, 326, 327, 329, 331, 337, 359, 362, 369, 399, 401, 405, 428, 431, 434, 436, 440, 445, 448, 463

Offset: 1

Author

Altug Alkan, Apr 04 2016

Keywords

• nonn

Comments

Integers k such that A000040(k) divides A003266(k).

If any Fibonacci(j) has a prime divisor p where p is the k-th prime for 1 <= j <= k, then k is a term of this sequence.

There are consecutive pairs such as (29, 30), (46, 47), (135, 136); what is the distribution of such pairs?

Examples

			5 is not a term because A000040(5) = 11 does not appear as a prime divisor of any of the first 5 nonzero Fibonacci numbers.
18 is a term because A003266(18) = 342696507457909818131702784000 is divisible by A000040(18) = 61.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003266, A270653.

Programs

PARI

lista(nn) = for(n=1, nn, my(p = prime(n)); if (lift(prod(i=1, n, Mod(fibonacci(i), p))) == 0, print1(n, ", ")));

Python

from sympy import prime
A271327_list = []
for n in range(1,10**3):
    p, a, b = prime(n), 1, 1
    for i in range(n):
        if not a:
            A271327_list.append(n)
            break
        a, b = b, (a + b) % p # Chai Wah Wu, Apr 08 2016