A230359 Prime numbers p such that their Fibonacci entry points are less than p+1.

5, 11, 13, 17, 19, 29, 31, 37, 41, 47, 53, 59, 61, 71, 73, 79, 89, 97, 101, 107, 109, 113, 131, 137, 139, 149, 151, 157, 173, 179, 181, 191, 193, 197, 199, 211, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 373, 379, 389, 397, 401, 409, 419, 421, 431, 433, 439, 449, 457, 461, 479, 491, 499

Offset: 1

Author

Brandon Avila and Tanya Khovanova, Oct 16 2013

Keywords

• nonn

Comments

For these primes p there exists a Fibonacci like sequence that doesn't contain multiples of p.

For other primes p the Fibonacci entry points are p+1. These primes are sequence A000057: Primes dividing all Fibonacci sequences.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• B. Avila and T. Khovanova, Free Fibonacci Sequences, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and J. Int. Seq. 17 (2014) # 14.8.5.

Crossrefs

A002144 is a subsequence.

Cf. A000057, A001177, A001602.

Programs

Maple

filter:= proc(n) local i,a,b,c;
  if not isprime(n) then return false fi;
  a:= 0; b:= 1;
  for i from 1 to n-1 do
   c:= b;
   b:= a+b mod n; if b = 0 then return true fi;
   a:= c;
  od;
false
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Sep 01 2020

Mathematica

A001177[n_] := For[k = 1, True, k++, If[Divisible[Fibonacci[k], n], Return[k]]]; A230359 = Reap[For[p = 2, p <= 499, p = NextPrime[p], If[A001177[p] < 1+p, Sow[p]]]][[2, 1]] (* Jean-François Alcover, Oct 21 2013 *)

Sage

def isA230359(p):
    return any(p.divides(fibonacci(k)) for k in (1..p))
print([p for p in primes(1, 500) if isA230359(p)]) # Peter Luschny, Nov 01 2019

Formula

{p in A000040: A001177(p) < 1+p}.