A309581 - OEIS

%I #41 Jun 25 2024 08:30:29

%S 3,11,17,19,43,59,67,73,83,89,97,107,113,131,139,163,179,193,211,227,

%T 233,241,251,257,281,283,307,331,337,347,379,401,419,433,443,449,467,

%U 491,499,523,547,563,571,577,587,601,617,619,641,643,659,673,683,691

%N Primes p with 2 zeros in a fundamental period of A000129 mod p.

%C Primes p such that A214027(p) = 2.

%C For p > 2, p is in this sequence if and only if 8 divides A175181(p), and if and only if 4 divides A214028(p). For a proof of the equivalence between A214027(p) = 2 and 4 dividing A214028(p), see Section 2 of my link below.

%C This sequence contains all primes congruent to 3 modulo 8. This corresponds to case (2) for k = 6 in the Conclusion of Section 1 of my link below.

%C Conjecturely, this sequence has density 5/12 in the primes. [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]

%H Jianing Song, <a href="/A309581/b309581.txt">Table of n, a(n) for n = 1..1600</a>

%H Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>

%o (PARI) forprime(p=2, 700, if(A214027(p)==2, print1(p, ", ")))

%Y Cf. A175181, A214028.

%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  | this seq | A309587

%Y Primes p such that w(p) = 4  | A053028  | A261580  | A309588

%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