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A281618 Fibonacci numbers F such that all the prime factors of F^2 + 1 are also Fibonacci numbers.

%I A281618 #20 Jan 28 2017 13:50:21
%S A281618 1,2,3,5,8,34,144,610,1134903170
%N A281618 Fibonacci numbers F such that all the prime factors of F^2 + 1 are also Fibonacci numbers.
%C A281618 The corresponding indices of F are 1 or 2, 3, 4, 5, 6, 9, 12, 15, 45, ... and A245236 is in this sequence.
%e A281618 a(9)^2+1 = Fibonacci(45)^2+1 = 1134903170^2+1 = 1288005205276048901 = 433494437 * 2971215073 = Fibonacci(43)*Fibonacci(47).
%p A281618 with(numtheory):with(combinat,fibonacci):nn:=100:
%p A281618 for n from 1 to nn do:
%p A281618   f:=fibonacci(n)^2+1:x:=factorset(f):n0:=nops(x):it:=0:
%p A281618     for m from 1 to n0 do:
%p A281618     c:=x[m]:
%p A281618     x1:=sqrt(5*c^2-4):x2:=sqrt(5*c^2+4):
%p A281618     if x1=floor(x1) or x2=floor(x2)
%p A281618      then
%p A281618      it:=it+1:
%p A281618      else
%p A281618     fi:
%p A281618 od:
%p A281618 if it=n0 then print(fibonacci(n)):else fi:od:
%t A281618 With[{s = Rest@ Fibonacci@ Range@ 120}, Select[s, Times @@ Boole@ Map[MemberQ[s, #] &, FactorInteger[#^2 + 1][[All, 1]]] > 0 &]] (* _Michael De Vlieger_, Jan 27 2017 *)
%o A281618 (PARI) isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
%o A281618 isokf(n) = {my(f = factor(fibonacci(n)^2+1)); for (k=1, #f~, if (!isfib(f[k,1]), return(0));); return(1);}
%o A281618 for (n=2, 50, if (isokf(n), print1(fibonacci(n), ", "))) \\ _Michel Marcus_, Jan 28 2017
%Y A281618 Cf. A000045, A245306, A245236.
%K A281618 nonn,more
%O A281618 1,2
%A A281618 _Michel Lagneau_, Jan 25 2017