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

1, 2, 3, 5, 8, 34, 144, 610, 1134903170

Offset: 1

Author

Michel Lagneau, Jan 25 2017

Keywords

• nonn
• more

Comments

The corresponding indices of F are 1 or 2, 3, 4, 5, 6, 9, 12, 15, 45, ... and A245236 is in this sequence.

Examples

			a(9)^2+1 = Fibonacci(45)^2+1 = 1134903170^2+1 = 1288005205276048901 = 433494437 * 2971215073 = Fibonacci(43)*Fibonacci(47).
		

Crossrefs

Cf. A000045, A245306, A245236.

Programs

Maple

with(numtheory):with(combinat,fibonacci):nn:=100:
for n from 1 to nn do:
  f:=fibonacci(n)^2+1:x:=factorset(f):n0:=nops(x):it:=0:
    for m from 1 to n0 do:
    c:=x[m]:
    x1:=sqrt(5*c^2-4):x2:=sqrt(5*c^2+4):
    if x1=floor(x1) or x2=floor(x2)
     then
     it:=it+1:
     else
    fi:
od:
if it=n0 then print(fibonacci(n)):else fi:od:

Mathematica

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 *)

PARI

isfib(n) = my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8));
isokf(n) = {my(f = factor(fibonacci(n)^2+1)); for (k=1, #f~, if (!isfib(f[k,1]), return(0));); return(1);}
for (n=2, 50, if (isokf(n), print1(fibonacci(n), ", "))) \\ Michel Marcus, Jan 28 2017