cp's OEIS Frontend

A309580 Primes p with 1 zero in a fundamental period of A000129 mod p.

%I A309580 #53 Jun 27 2025 21:00:09
%S A309580 2,7,23,31,41,47,71,79,103,127,151,167,191,199,223,239,263,271,311,
%T A309580 313,353,359,367,383,409,431,439,457,463,479,487,503,599,607,631,647,
%U A309580 719,727,743,751,761,809,823,839,863,887,911,919,967,983,991,1031,1039,1063,1087,1103,1129,1151,1201,1223,1231,1279
%N A309580 Primes p with 1 zero in a fundamental period of A000129 mod p.
%C A309580 Primes p such that A214027(p) = 1.
%C A309580 For p > 2, p is in this sequence if and only if A175181(p) == 2 (mod 4), and if and only if A214028(p) == 2 (mod 4). For a proof of the equivalence between A214027(p) = 1 and A214028(p) == 2 (mod 4), see Section 2 of my link below.
%C A309580 This sequence contains all primes congruent to 7 modulo 8. This corresponds to case (3) for k = 6 in the Conclusion of Section 1 of my link below.
%C A309580 Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 7/24 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 A309580 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 A309580 Jianing Song, <a href="/A309580/b309580.txt">Table of n, a(n) for n = 1..1280</a>
%H A309580 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%o A309580 (PARI) forprime(p=2, 1300, if(A214027(p)==1, print1(p, ", ")))
%Y A309580 Cf. A175181, A214028.
%Y A309580 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 A309580                              |   m=1    |   m=2    |   m=3
%Y A309580 -----------------------------+----------+----------+---------
%Y A309580 The sequence {x(n)}          | A000045  | A000129  | A006190
%Y A309580 The sequence {w(k)}          | A001176  | A214027  | A322906
%Y A309580 Primes p such that w(p) = 1  | A112860* | this seq | A309586
%Y A309580 Primes p such that w(p) = 2  | A053027  | A309581  | A309587
%Y A309580 Primes p such that w(p) = 4  | A053028  | A261580  | A309588
%Y A309580 Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
%Y A309580 Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
%Y A309580 Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
%Y A309580 * and also A053032 U {2}
%K A309580 nonn
%O A309580 1,1
%A A309580 _Jianing Song_, Aug 10 2019