cp's OEIS Frontend

A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.

%I A309587 #33 Jun 25 2024 08:30:03
%S A309587 7,11,17,19,31,47,59,67,71,83,113,151,163,167,223,227,239,257,271,307,
%T A309587 313,331,337,359,379,383,431,463,479,487,499,521,587,601,619,631,641,
%U A309587 643,673,683,691,739,743,787,809,811,827,839,863,947,967,983
%N A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.
%C A309587 Primes p such that A322906(p) = 2.
%C A309587 For p > 2, p is in this sequence if and only if 8 divides A175182(p), and if and only if 4 divides A322907(p). For a proof of the equivalence between A322906(p) = 2 and 4 dividing A322907(p), see Section 2 of my link below.
%C A309587 This sequence contains all primes congruent to 7, 11, 15, 19, 31, 47 modulo 52. This corresponds to case (2) for k = 11 in the Conclusion of Section 1 of my link below.
%C A309587 Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H A309587 Jianing Song, <a href="/A309587/b309587.txt">Table of n, a(n) for n = 1..1200</a>
%H A309587 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%o A309587 (PARI) forprime(p=2, 1000, if(A322906(p)==2, print1(p, ", ")))
%Y A309587 Cf. A006190, A175182, A322907.
%Y A309587 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 A309587                              |   m=1    |   m=2   |   m=3
%Y A309587 -----------------------------+----------+---------+----------
%Y A309587 The sequence {x(n)}          | A000045  | A000129 | A006190
%Y A309587 The sequence {w(k)}          | A001176  | A214027 | A322906
%Y A309587 Primes p such that w(p) = 1  | A112860* | A309580 | A309586
%Y A309587 Primes p such that w(p) = 2  | A053027  | A309581 | this seq
%Y A309587 Primes p such that w(p) = 4  | A053028  | A261580 | A309588
%Y A309587 Numbers k such that w(k) = 1 | A053031  | A309583 | A309591
%Y A309587 Numbers k such that w(k) = 2 | A053030  | A309584 | A309592
%Y A309587 Numbers k such that w(k) = 4 | A053029  | A309585 | A309593
%Y A309587 * and also A053032 U {2}
%K A309587 nonn
%O A309587 1,1
%A A309587 _Jianing Song_, Aug 10 2019