%I #10 Mar 22 2017 03:39:12

%S 3,21,55,377,987,6765,17711,121393,317811,2178309,5702887,39088169,

%T 102334155,701408733,1836311903,12586269025,32951280099,225851433717,

%U 591286729879,4052739537881,10610209857723,72723460248141,190392490709135,1304969544928657,3416454622906707

%N 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.

%C 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

%C 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;

%C 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).

%e F = 3 has the proper Fibonacci divisor G=1, and F/G = 3 is a Lucas number.

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

%Y Cf. A000045, A000032, A014437, A014447.

%K nonn

%O 1,1

%A _Vladimir Shevelev_, Oct 07 2010