cp's OEIS Frontend

A309586 Primes p with 1 zero in a fundamental period of A006190 mod p.

%I A309586 #33 Jun 25 2024 08:29:58
%S A309586 2,3,23,43,53,61,79,101,103,107,127,131,139,173,179,191,199,211,251,
%T A309586 263,277,283,311,347,367,419,433,439,443,467,491,503,523,547,563,569,
%U A309586 571,599,607,647,659,677,719,727,751,757,823,829,859,881,883,887,907
%N A309586 Primes p with 1 zero in a fundamental period of A006190 mod p.
%C A309586 Primes p such that A322906(p) = 1.
%C A309586 For p > 2, p is in this sequence if and only if A175182(p) == 2 (mod 4), and if and only if A322907(p) == 2 (mod 4). For a proof of the equivalence between A322906(p) = 1 and A322907(p) == 2 (mod 4), see Section 2 of my link below.
%C A309586 This sequence contains all primes congruent to 3, 23, 27, 35, 43, 51 modulo 52. This corresponds to case (3) for k = 11 in the Conclusion of Section 1 of my link below.
%C A309586 Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H A309586 Jianing Song, <a href="/A309586/b309586.txt">Table of n, a(n) for n = 1..1200</a>
%H A309586 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%o A309586 (PARI) forprime(p=2, 900, if(A322906(p)==1, print1(p, ", ")))
%Y A309586 Cf. A175182, A322907.
%Y A309586 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 A309586                              |   m=1    |   m=2   |   m=3
%Y A309586 -----------------------------+----------+---------+----------
%Y A309586 The sequence {x(n)}          | A000045  | A000129 | A006190
%Y A309586 The sequence {w(k)}          | A001176  | A214027 | A322906
%Y A309586 Primes p such that w(p) = 1  | A112860* | A309580 | this seq
%Y A309586 Primes p such that w(p) = 2  | A053027  | A309581 | A309587
%Y A309586 Primes p such that w(p) = 4  | A053028  | A261580 | A309588
%Y A309586 Numbers k such that w(k) = 1 | A053031  | A309583 | A309591
%Y A309586 Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
%Y A309586 Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
%Y A309586 * and also A053032 U {2}
%K A309586 nonn
%O A309586 1,1
%A A309586 _Jianing Song_, Aug 10 2019