A272575 Perfect powers that are the sum of two Fibonacci numbers.

1, 4, 8, 9, 16, 36, 144, 1000, 1600, 14930496

Offset: 1

Author

Altug Alkan, May 03 2016

Keywords

• nonn

Comments

Intersection of A001597 and A084176.

Listed terms are 1, 2^2, 2^3, 3^2, 2^4, 6^2, 12^2, 10^3, 40^2, 3864^2.

First five terms are also members of A000961.

Conjecture: there are no more terms in this sequence. Any remaining terms must have over 10000 digits. - Charles R Greathouse IV, May 04 2016

Examples

			8 is a term because 2^3 = 3 + 5.
		

Crossrefs

Cf. A000045, A001597, A084176, A111378.

Programs

Mathematica

Select[Range[10^4], Function[k, Or[k == 1, GCD @@ Map[Last, FactorInteger@ k] > 1] && Total@ Map[Times @@ Boole@ Map[MemberQ[s, #] &, #] &, Transpose@ {#, k - #} &@ Range[0, Floor[k/2]]] > 0]] (* Michael De Vlieger, May 03 2016 *)

PARI

list(lim)=my(upper=log(lim*sqrt(5))\log((1+sqrt(5))/2)+1, t, tt, v=List([1])); if(fibonacci(t)>lim, t--); for(i=3, upper, t=fibonacci(i); for(j=2, i-1, tt=t+fibonacci(j); if(tt>lim, break); if(ispower(tt), listput(v, tt)))); Set(v) \\ Charles R Greathouse IV, May 03 2016