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A272712
Perfect powers that are the difference of two nonnegative Fibonacci numbers.
0
1, 4, 8, 16, 32, 81, 144, 225, 343, 576
OFFSET
1,2
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
EXAMPLE
32 is a term because 32 = 2^5 = 34 - 2 = Fibonacci(9) - Fibonacci(3).
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 *)
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Altug Alkan, May 05 2016
STATUS
approved