cp's OEIS Frontend

A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p.

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