A275774 Numbers m with the property that, when a and b are positive integers such that a*b = m, |a-b| is a Fibonacci number.

1, 2, 3, 6, 14, 35, 611, 3524579

Offset

1,2

Comments

a(n) is of the form Fibonacci(k) + 1. Is this sequence finite?

There are probably no more terms. If a(9) exists, it is greater than 10^200. - Charles R Greathouse IV, Aug 08 2016

Links

Table of n, a(n) for n=1..8.

Example

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.

Mathematica

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}]

Prog

(PARI) isFibonacci(n)=my(k=n^2); k+=((k+1)<<2); issquare(k)||(n>0&&issquare(k-8))
is(n)=fordiv(n, d, if(!isFibonacci(abs(n/d-d)), return(0))); 1 \\ Charles R Greathouse IV, Aug 08 2016

Crossrefs

Cf. A000045, A038548.

Sequence in context: A093467 A246640 A080408 * A319323 A341630 A001420

Adjacent sequences: A275771 A275772 A275773 * A275775 A275776 A275777

Keyword

nonn

Author

Michel Lagneau, Aug 08 2016

Extensions

a(1) inserted by Charles R Greathouse IV, Aug 08 2016

Status

approved