cp's OEIS Frontend

A112860 2 together with A053032.

%I A112860 #29 Jan 05 2025 19:51:38
%S A112860 2,11,19,29,31,59,71,79,101,131,139,151,179,181,191,199,211,229,239,
%T A112860 251,271,311,331,349,359,379,419,431,439,461,479,491,499,509,521,541,
%U A112860 571,599,619,631,659,691,709,719,739,751,809,811,839,859,911,919,941,971
%N A112860 2 together with A053032.
%C A112860 Consists of the primes that are in neither A053027 nor A053028.
%C A112860 From _Jianing Song_, Jun 16 2024: (Start)
%C A112860 Primes p such that A001176(p) = 1.
%C A112860 For p > 2, p is in this sequence if and only if A001175(p) == 2 (mod 4), and if and only if A001177(p) == 2 (mod 4). For a proof of the equivalence between A001176(p) = 1 and A001177(p) == 2 (mod 4), see Section 2 of my link below.
%C A112860 This sequence contains all primes congruent to 11, 19 (mod 20). This corresponds to case (3) for k = 3 in the Conclusion of Section 1 of my link below.
%C A112860 Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H A112860 T. D. Noe, <a href="/A112860/b112860.txt">Table of n, a(n) for n=1..1001</a>
%H A112860 C. Ballot and M. Elia, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-1/quartballot01_2007.pdf">Rank and period of primes in the Fibonacci sequence; a trichotomy</a>, Fib. Quart., 45 (No. 1, 2007), 56-63 (The sequence B1).
%H A112860 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%Y A112860 Cf. A001175, A001177.
%Y A112860 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 A112860                              |   m=1     |   m=2   |   m=3
%Y A112860 -----------------------------+-----------+---------+---------
%Y A112860 The sequence {x(n)}          | A000045   | A000129 | A006190
%Y A112860 The sequence {w(k)}          | A001176   | A214027 | A322906
%Y A112860 Primes p such that w(p) = 1  | this seq* | A309580 | A309586
%Y A112860 Primes p such that w(p) = 2  | A053027   | A309581 | A309587
%Y A112860 Primes p such that w(p) = 4  | A053028   | A261580 | A309588
%Y A112860 Numbers k such that w(k) = 1 | A053031   | A309583 | A309591
%Y A112860 Numbers k such that w(k) = 2 | A053030   | A309584 | A309592
%Y A112860 Numbers k such that w(k) = 4 | A053029   | A309585 | A309593
%Y A112860 * and also A053032 U {2}
%K A112860 nonn
%O A112860 1,1
%A A112860 _N. J. A. Sloane_, Nov 30 2007