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%I #8 Aug 09 2016 17:25:15
%S 1,2,3,6,14,35,611,3524579
%N Numbers m with the property that, when a and b are positive integers such that a*b = m, |a-b| is a Fibonacci number.
%C a(n) is of the form Fibonacci(k) + 1. Is this sequence finite?
%C There are probably no more terms. If a(9) exists, it is greater than 10^200. - _Charles R Greathouse IV_, Aug 08 2016
%e 35 is in the sequence because A038548(35) = 2 => two decompositions of 35 = 1*35 = 5*7 => 35-1 = 34 and 7-2 = 5 are Fibonacci numbers.
%t Do[ds=Divisors[n];If[EvenQ[Length[ds]],ok=True;k=1;While[k<=Length[ds]/2&&(ok=IntegerQ[Sqrt[5*(ds[[k]]-ds[[-k]])^2+4]]||IntegerQ[Sqrt[5*(ds[[k]]-ds[[-k]])^2-4]]),k++];If[ok,Print[n]]],{n,2,10^7}]
%o (PARI) isFibonacci(n)=my(k=n^2);k+=((k+1)<<2);issquare(k)||(n>0&&issquare(k-8))
%o is(n)=fordiv(n, d, if(!isFibonacci(abs(n/d-d)), return(0))); 1 \\ _Charles R Greathouse IV_, Aug 08 2016
%Y Cf. A000045, A038548.
%K nonn
%O 1,2
%A _Michel Lagneau_, Aug 08 2016
%E a(1) inserted by _Charles R Greathouse IV_, Aug 08 2016
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