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

%I A230359 #31 Jan 11 2022 22:11:09
%S A230359 5,11,13,17,19,29,31,37,41,47,53,59,61,71,73,79,89,97,101,107,109,113,
%T A230359 131,137,139,149,151,157,173,179,181,191,193,197,199,211,229,233,239,
%U A230359 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
%N A230359 Prime numbers p such that their Fibonacci entry points are less than p+1.
%C A230359 For these primes p there exists a Fibonacci like sequence that doesn't contain multiples of p.
%C A230359 For other primes p the Fibonacci entry points are p+1. These primes are sequence A000057: Primes dividing all Fibonacci sequences.
%H A230359 Robert Israel, <a href="/A230359/b230359.txt">Table of n, a(n) for n = 1..10000</a>
%H A230359 B. Avila and T. Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>.
%F A230359 {p in A000040: A001177(p) < 1+p}.
%p A230359 filter:= proc(n) local i,a,b,c;
%p A230359   if not isprime(n) then return false fi;
%p A230359   a:= 0; b:= 1;
%p A230359   for i from 1 to n-1 do
%p A230359    c:= b;
%p A230359    b:= a+b mod n; if b = 0 then return true fi;
%p A230359    a:= c;
%p A230359   od;
%p A230359 false
%p A230359 end proc:
%p A230359 select(filter, [seq(i,i=3..1000,2)]); # _Robert Israel_, Sep 01 2020
%t A230359 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 *)
%o A230359 (Sage)
%o A230359 def isA230359(p):
%o A230359     return any(p.divides(fibonacci(k)) for k in (1..p))
%o A230359 print([p for p in primes(1, 500) if isA230359(p)]) # _Peter Luschny_, Nov 01 2019
%Y A230359 A002144 is a subsequence.
%Y A230359 Cf. A000057, A001177, A001602.
%K A230359 nonn
%O A230359 1,1
%A A230359 _Brandon Avila_ and _Tanya Khovanova_, Oct 16 2013