A272712 Perfect powers that are the difference of two nonnegative Fibonacci numbers.

1, 4, 8, 16, 32, 81, 144, 225, 343, 576

Offset: 1

Author

Altug Alkan, May 05 2016

Keywords

• nonn
• more

Comments

Listed 10 terms are 1, 2^2, 2^3, 2^4, 2^5, 3^4, 12^2, 15^2, 3^5, 24^2.

1, 4, 8, 16, 32, 81, 343 are also members of A000961.

1, 4, 8, 16, 144 are in the intersection of this sequence and A272575.

Is this sequence finite?

If a(11) exists, it must be larger than 10^2000. - Giovanni Resta, May 25 2016

Examples

			32 is a term because 32 = 2^5 = 34 - 2 = Fibonacci(9) - Fibonacci(3).
		

Crossrefs

Cf. A000961, A007298, A001597, A219114, A272575.

Programs

Maple

isA272712 := proc(n)
    isA001597(n) and isA007298(n) ; #uses code in A001597 and A007298
end proc:
for n from 1 do
    if isA272712(n) then
        printf("%d\n",n) ;
    end if;
end do: # R. J. Mathar, May 25 2016

Mathematica

isA001597[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1;
isA007298[n_] := Module[{i, Fi, j, Fj}, For[i = 0, True, i++, Fi = Fibonacci[i]; For[j = i, True, j++, Fj = Fibonacci[j]; Which[Fj - Fi == n, Return@True, Fj - Fi > n, Break[]]]; Fj := Fibonacci[i + 1]; If[Fj - Fi > n, Return@False]]];
Select[Range[1000], isA001597[#] && isA007298[#]&] (* Jean-François Alcover, Nov 16 2023, after R. J. Mathar in A007298 *)