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%I #14 May 04 2016 20:53:25
%S 1,4,8,9,16,36,144,1000,1600,14930496
%N Perfect powers that are the sum of two Fibonacci numbers.
%C Intersection of A001597 and A084176.
%C Listed terms are 1, 2^2, 2^3, 3^2, 2^4, 6^2, 12^2, 10^3, 40^2, 3864^2.
%C First five terms are also members of A000961.
%C Conjecture: there are no more terms in this sequence. Any remaining terms must have over 10000 digits. - _Charles R Greathouse IV_, May 04 2016
%e 8 is a term because 2^3 = 3 + 5.
%t 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 *)
%o (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
%Y Cf. A000045, A001597, A084176, A111378.
%K nonn
%O 1,2
%A _Altug Alkan_, May 03 2016
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