%I #32 Jun 25 2024 08:30:07
%S 5,13,29,37,41,73,89,97,109,137,149,157,181,193,197,229,233,241,269,
%T 281,293,317,349,353,373,389,397,401,409,421,449,457,461,509,541,557,
%U 577,593,613,617,653,661,701,709,733,761,769,773,797,821,853,857,877
%N Primes p with 4 zeros in a fundamental period of A006190 mod p.
%C Primes p such that A322906(p) = 4.
%C For p > 2, p is in this sequence if and only if A175182(p) == 4 (mod 8), and if and only if A322907(p) is odd. For a proof of the equivalence between A322906(p) = 4 and A322907(p) being odd, see Section 2 of my link below.
%C This sequence contains all primes congruent to 5, 21, 33, 37, 41, 45 modulo 52. This corresponds to case (1) for k = 11 in the Conclusion of Section 1 of my link below.
%C Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H Jianing Song, <a href="/A309588/b309588.txt">Table of n, a(n) for n = 1..1200</a>
%H Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%o (PARI) forprime(p=2, 900, if(A322906(p)==4, print1(p, ", ")))
%Y Cf. A006190, A175182, A322907.
%Y Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
%Y | m=1 | m=2 | m=3
%Y -----------------------------+----------+---------+----------
%Y The sequence {x(n)} | A000045 | A000129 | A006190
%Y The sequence {w(k)} | A001176 | A214027 | A322906
%Y Primes p such that w(p) = 1 | A112860* | A309580 | A309586
%Y Primes p such that w(p) = 2 | A053027 | A309581 | A309587
%Y Primes p such that w(p) = 4 | A053028 | A261580 | this seq
%Y Numbers k such that w(k) = 1 | A053031 | A309583 | A309591
%Y Numbers k such that w(k) = 2 | A053030 | A309584 | A309592
%Y Numbers k such that w(k) = 4 | A053029 | A309585 | A309593
%Y * and also A053032 U {2}
%K nonn
%O 1,1
%A _Jianing Song_, Aug 10 2019