[seqfan] Re: Consecutive composite Fibonacci numbers

[Richard Mathar]

Referring to http://list.seqfan.eu/pipermail/seqfan/2010-November/006476.html

njas> Date: Thu, 18 Nov 2010 22:31:05 -0500
njas> To: seqfan at list.seqfan.eu
njas> Subject: [seqfan] Re: Consecutive composite Fibonacci numbers
njas> 
njas> if we ask for smallest Fib which is the start of at least
njas> n consecutive composite Fibonacci nums, we get 8,21,21,2584,2584,...
njas> (subject to checking)

With offset 1:

Smallest Fibonacci number at the start of at least n consecutive composite
Fibonacci numbers

8, 21, 21, 2584, 2584, 832040, 832040, 832040, 832040, 832040, 832040, 832040,
    832040, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976,
    4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976,
    4807526976, 4807526976, 4807526976, 4807526976, 4807526976, 4807526976,
    4807526976, 4807526976, 4807526976, 4807526976, 4807526976,
    160500643816367088, 160500643816367088, 160500643816367088,
    160500643816367088, 160500643816367088, 160500643816367088,
    160500643816367088, 160500643816367088, 160500643816367088,
    160500643816367088, 160500643816367088, 160500643816367088,
    30960598847965113057878492344, 30960598847965113057878492344,
    30960598847965113057878492344

min{ A000045(i): A000045(j) in A002808 forall j=i..i+n-1}

Associated indices:
6, 8, 8, 18, 18, 30, 30, 30, 30, 30, 30, 30, 30, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48,
 48, 48, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 84, 138, 138, 138


njas> if we ask for smallest Fib which is the start of exactly
njas> n consecutive composite Fibonacci nums, we get 8,?,21,?,...

8, 34, 21, 4181, 2584, 24157817, 14930352, 9227465, 5702887, 3524578, 2178309, 1346269, 832040, 117669030460994, 72723460248141,
    44945570212853, 27777890035288, 17167680177565, 10610209857723, 6557470319842, 4052739537881, 2504730781961, 1548008755920,
    956722026041, 591286729879, 365435296162, 225851433717, 139583862445, 86267571272, 53316291173, 32951280099, 20365011074,
    12586269025, 7778742049, 4807526976, 31940434634990099905, 19740274219868223167, 12200160415121876738, 7540113804746346429,
    4660046610375530309

Associated indices in A000045:
6, 9, 8, 19, 18, 37, 36, 35, 34, 33, 32, 31, 30, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50,
 49, 48, 95, 94, 93, 92, 91

Maple program for the latter, returning the indices:

A := proc(n)
        for i from 6 do
                allc := 0;
                for k from 0 do
                        if not isprime( combinat[fibonacci](i+k)) then
                                allc := allc+1;
                        else
                                break;
                        end if;
                end do:
                if allc =n then
                        return i ;
                end if;
        end do:
end proc:
seq(A(n),n=1..40) ;

A more useful definition which avoids taking a Fibonacci number from
within a sequence of consecutive Fibonacci composites is probably

Smallest Fibonacci number, following a Fibonacci prime, at the start
of at least n consecutive composite Fibonacci numbers; 0 if this does
not exist.

which essentially looks for Fibonacci prime numbers
and can be derived from A001605 and A005478.

I have not submitted any of these sequences.

Richard Mathar