%I #30 Nov 16 2023 15:56:46

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

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

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

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

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

%C Is this sequence finite?

%C If a(11) exists, it must be larger than 10^2000. - _Giovanni Resta_, May 25 2016

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

%p isA272712 := proc(n)

%p isA001597(n) and isA007298(n) ; #uses code in A001597 and A007298

%p end proc:

%p for n from 1 do

%p if isA272712(n) then

%p printf("%d\n",n) ;

%p end if;

%p end do: # _R. J. Mathar_, May 25 2016

%t isA001597[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1;

%t 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]]];

%t Select[Range[1000], isA001597[#] && isA007298[#]&] (* _Jean-François Alcover_, Nov 16 2023, after _R. J. Mathar_ in A007298 *)

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

%K nonn,more

%O 1,2

%A _Altug Alkan_, May 05 2016