|
%I #27 Jun 25 2024 08:30:12
%S 2,11,19,29,31,59,71,79,101,131,139,151,179,181,191,199,211,229,239,
%T 251,271,311,331,349,359,379,419,431,439,461,479,491,499,509,521,541,
%U 571,599,619,631,659,691,709,719,739,751,809,811,839,859,911,919,941,971
%N 2 together with A053032.
%C Consists of the primes that are in neither A053027 nor A053028.
%C From _Jianing Song_, Jun 16 2024: (Start)
%C Primes p such that A001176(p) = 1.
%C 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 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 Conjecturely, this sequence has density 1/3 in the primes. (End) [Comment rewritten by _Jianing Song_, Jun 16 2024 and Jun 25 2024]
%H T. D. Noe, <a href="/A112860/b112860.txt">Table of n, a(n) for n=1..1001</a>
%H C. Ballot and M. Elia, <a href="http://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 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%Y Cf. A001175, A001177.
%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 | this seq* | A309580 | A309586
%Y Primes p such that w(p) = 2 | A053027 | A309581 | 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,changed
%O 1,1
%A _N. J. A. Sloane_, Nov 30 2007
|
