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A281618 Fibonacci numbers F such that all the prime factors of F^2 + 1 are also Fibonacci numbers. 0
1, 2, 3, 5, 8, 34, 144, 610, 1134903170 (list; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=1..9.

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

Cf. A000045, A245306, A245236.

Sequence in context: A275524 A177195 A178355 * A059359 A042069 A041008

Adjacent sequences:  A281615 A281616 A281617 * A281619 A281620 A281621

KEYWORD

nonn,more

AUTHOR

Michel Lagneau, Jan 25 2017

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)