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A281618
Fibonacci numbers F such that all the prime factors of F^2 + 1 are also Fibonacci numbers.
3
1, 2, 3, 5, 8, 34, 144, 610, 1134903170
OFFSET
1,2
COMMENTS
The corresponding indices of F are 1 or 2, 3, 4, 5, 6, 9, 12, 15, 45, ... and A245236 is in this sequence.
EXAMPLE
a(9)^2+1 = Fibonacci(45)^2+1 = 1134903170^2+1 = 1288005205276048901 = 433494437 * 2971215073 = Fibonacci(43)*Fibonacci(47).
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Michel Lagneau, Jan 25 2017
STATUS
approved