A181156 Odd Fibonacci numbers F which have a proper Fibonacci divisor G such that F/G is a Lucas number or a product of Lucas numbers.

3, 21, 55, 377, 987, 6765, 17711, 121393, 317811, 2178309, 5702887, 39088169, 102334155, 701408733, 1836311903, 12586269025, 32951280099, 225851433717, 591286729879, 4052739537881, 10610209857723, 72723460248141, 190392490709135, 1304969544928657, 3416454622906707

Offset: 1

Author

Vladimir Shevelev, Oct 07 2010

Keywords

• nonn

Comments

A conjectural statement that, for odd prime p, the ratio F_{p^2}/F_{p} is never a Lucas number or a product of some Lucas numbers, yields that

a) an odd Fibonacci number F is in the sequence iff for its maximal proper Fibonacci divisor G, we have: ind G does not equal sqrt(ind F) and F/G does not have a proper Fibonacci divisor > 3;

b) an odd Fibonacci number F is in the sequence iff its index has one of the forms: 6k+2 or 6k+4 (see A047235).

Examples

			F = 3 has the proper Fibonacci divisor G=1, and F/G = 3 is a Lucas number.
F = 317811 has the proper Fibonacci divisors 3, 13, and 377, and F/377 = 843 is a Lucas number.
		

Crossrefs

Cf. A000045, A000032, A014437, A014447.