A338794 Indices k of Fibonacci numbers F(k) such that F(k)^2 + 1 has no Fibonacci prime factor.

39, 60, 69, 72, 99, 102, 105, 108, 111, 150, 165, 180, 192, 195, 198, 225, 228, 231, 240, 270, 279, 282, 309, 312, 315, 348, 351, 381, 399, 420, 441, 459, 462, 465, 489, 501, 522, 588, 591, 600, 615, 618, 642, 645, 660, 675, 702, 741, 759, 771, 810, 822, 825, 828

Offset

1,1

Comments

Numbers k such that A338762(k) = 0.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Example

39 is in the sequence because F(39)^2 + 1 = 63245986^2 + 1 = 73*149*2221*2789*59369 with no Fibonacci prime factors.
38 is not in the sequence because F(38)^2 + 1 = 39088169^2 + 1 =  2*73*149*233*2221*135721. The numbers and 2, 233 are Fibonacci prime factors.

Maple

a:= proc(n) local F, m, t; F, m, t:=
      [1, 2], 0, (<<0|1>, <1|1>>^n)[2, 1]^2+1;
      while F[2]<=t do if isprime(F[2]) and irem(t, F[2])=0
        then m:=F[2] fi; F:= [F[2], F[1]+F[2]]
      od; m
    end:
for n from 1 to 100 do :
if a(n)=0 then printf(`%d, `, n):else fi:
od: # program from Alois P. Heinz, adapted for the sequence. See A338762.

Mathematica

A338762[n_] := Module[{F, m, t}, F = {1, 2}; m = 0; t = MatrixPower[{{0, 1}, {1, 1}}, n][[2, 1]]^2 + 1; While[F[[2]] <= t, If[PrimeQ[F[[2]]] && Mod[t, F[[2]]] == 0, m = F[[2]]]; F = {F[[2]], F[[1]] + F[[2]]}]; m];
Reap[For[k = 1, k <= 1000, k++, If[A338762[k] == 0, Print[k]; Sow[k]]]][[2, 1]] (* Jean-François Alcover, Mar 16 2025, after Alois P. Heinz *)

Prog

(PARI) isok(n) = {my(i=0, f=0, x=fibonacci(n)^2+1, m=0); while(f < x, i++; f = fibonacci(i); if (ispseudoprime(f) && (x%f) == 0, return (0)); ); return(1); } \\ Michel Marcus, Nov 13 2020

Crossrefs

Cf. A000045, A005478, A168063, A245306, A338762.

Sequence in context: A168530 A253436 A253429 * A050780 A039353 A043176

Adjacent sequences: A338791 A338792 A338793 * A338795 A338796 A338797

Keyword

nonn

Author

Michel Lagneau, Nov 09 2020

Status

approved