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

2, 7, 23, 31, 41, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 313, 353, 359, 367, 383, 409, 431, 439, 457, 463, 479, 487, 503, 599, 607, 631, 647, 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

Offset

1,1

Comments

Primes p such that A214027(p) = 1.

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.

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.

Conjecturely, this sequence has density 7/24 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]

Links

Jianing Song, Table of n, a(n) for n = 1..1280

Jianing Song, Lucas sequences and entry point modulo p

Prog

(PARI) forprime(p=2, 1300, if(A214027(p)==1, print1(p, ", ")))

Crossrefs

Cf. A175181, A214028.

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.

| m=1 | m=2 | m=3

-----------------------------+----------+----------+---------

The sequence {x(n)} | A000045 | A000129 | A006190

The sequence {w(k)} | A001176 | A214027 | A322906

Primes p such that w(p) = 1 | A112860* | this seq | A309586

Primes p such that w(p) = 2 | A053027 | A309581 | A309587

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

Numbers k such that w(k) = 1 | A053031 | A309583 | A309591

Numbers k such that w(k) = 2 | A053030 | A309584 | A309592

Numbers k such that w(k) = 4 | A053029 | A309585 | A309593

* and also A053032 U {2}

Sequence in context: A045381 A180537 A042145 * A186098 A040098 A045315

Adjacent sequences: A309577 A309578 A309579 * A309581 A309582 A309583

Keyword

nonn

Author

Jianing Song, Aug 10 2019

Status

approved