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

%I A272712 #30 Nov 16 2023 15:56:46
%S A272712 1,4,8,16,32,81,144,225,343,576
%N A272712 Perfect powers that are the difference of two nonnegative Fibonacci numbers.
%C A272712 Listed 10 terms are 1, 2^2, 2^3, 2^4, 2^5, 3^4, 12^2, 15^2, 3^5, 24^2.
%C A272712 1, 4, 8, 16, 32, 81, 343 are also members of A000961.
%C A272712 1, 4, 8, 16, 144 are in the intersection of this sequence and A272575.
%C A272712 Is this sequence finite?
%C A272712 If a(11) exists, it must be larger than 10^2000. - _Giovanni Resta_, May 25 2016
%e A272712 32 is a term because 32 = 2^5 = 34 - 2 = Fibonacci(9) - Fibonacci(3).
%p A272712 isA272712 := proc(n)
%p A272712     isA001597(n) and isA007298(n) ; #uses code in A001597 and A007298
%p A272712 end proc:
%p A272712 for n from 1 do
%p A272712     if isA272712(n) then
%p A272712         printf("%d\n",n) ;
%p A272712     end if;
%p A272712 end do: # _R. J. Mathar_, May 25 2016
%t A272712 isA001597[n_] := n == 1 || GCD @@ FactorInteger[n][[All, 2]] > 1;
%t A272712 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 A272712 Select[Range[1000], isA001597[#] && isA007298[#]&] (* _Jean-François Alcover_, Nov 16 2023, after _R. J. Mathar_ in A007298 *)
%Y A272712 Cf. A000961, A007298, A001597, A219114, A272575.
%K A272712 nonn,more
%O A272712 1,2
%A A272712 _Altug Alkan_, May 05 2016