cp's OEIS Frontend

A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p.

%I A309581 #47 Jun 27 2025 21:00:18
%S A309581 3,11,17,19,43,59,67,73,83,89,97,107,113,131,139,163,179,193,211,227,
%T A309581 233,241,251,257,281,283,307,331,337,347,379,401,419,433,443,449,467,
%U A309581 491,499,523,547,563,571,577,587,601,617,619,641,643,659,673,683,691
%N A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p.
%C A309581 Primes p such that A214027(p) = 2.
%C A309581 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 A309581 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 A309581 Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 5/12 in the primes; see the end of Section 1 of my link. [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%C A309581 The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - _Jianing Song_, Jun 26 2025
%H A309581 Jianing Song, <a href="/A309581/b309581.txt">Table of n, a(n) for n = 1..1600</a>
%H A309581 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%o A309581 (PARI) forprime(p=2, 700, if(A214027(p)==2, print1(p, ", ")))
%Y A309581 Cf. A175181, A214028.
%Y A309581 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 A309581                              |   m=1    |   m=2    |   m=3
%Y A309581 -----------------------------+----------+----------+---------
%Y A309581 The sequence {x(n)}          | A000045  | A000129  | A006190
%Y A309581 The sequence {w(k)}          | A001176  | A214027  | A322906
%Y A309581 Primes p such that w(p) = 1  | A112860* | A309580  | A309586
%Y A309581 Primes p such that w(p) = 2  | A053027  | this seq | A309587
%Y A309581 Primes p such that w(p) = 4  | A053028  | A261580  | A309588
%Y A309581 Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
%Y A309581 Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
%Y A309581 Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
%Y A309581 * and also A053032 U {2}
%K A309581 nonn
%O A309581 1,1
%A A309581 _Jianing Song_, Aug 10 2019