A309587 Primes p with 2 zeros in a fundamental period of A006190 mod p.

7, 11, 17, 19, 31, 47, 59, 67, 71, 83, 113, 151, 163, 167, 223, 227, 239, 257, 271, 307, 313, 331, 337, 359, 379, 383, 431, 463, 479, 487, 499, 521, 587, 601, 619, 631, 641, 643, 673, 683, 691, 739, 743, 787, 809, 811, 827, 839, 863, 947, 967, 983

Offset

1,1

Comments

Primes p such that A322906(p) = 2.

For p > 2, p is in this sequence if and only if (all these conditions are equivalent):

(a) 8 divides A175182(p);

(b) 8 divides ord(p,(3+sqrt(13))/2), where ord(p,u) is the smallest integer k > 0 such that (u^k - 1)/p is an algebraic integer;

(c) 4 divides ord(p,(11+3*sqrt(13))/2);

(d) 4 divides A322907(p);

(e) 4 divides ord(p,-(11+3*sqrt(13))/2).

In general, let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let pi(k) be the Pisano period of {x(n)} modulo k, i.e., pi(k) = min{l > 0 : x(n+l) == x(n) (mod k) for all n}, r(k) = min{l > 0 : k divides x(l)} and w(k) be the number of zeros in a fundamental period of {x(n)} modulo k. Let u = (m + sqrt(m^2+4))/2, p be an odd prime, then these conditions are equivalent:

(1) w(p) = 2;

(2) 8 divides pi(p);

(3) 8 divides ord(p,u);

(4) 4 divides ord(p,u^2);

(5) 4 divides r(p);

(6) 4 divides ord(p,-u^2).

This can be shown by noting that pi(p) = p^c*ord(p,u) and r(p) = p^c*ord(p,-u^2) for some c (if p does not divide m^2 + 4 then c = 0, otherwise c = 1). Also, Pi(p) is always even, so ord(p,u) is always even.

This sequence contains all primes congruent to 7, 11, 15, 19, 31, 47 modulo 52.

Conjecturely, this sequence has density 1/3 in the primes.

Links

Jianing Song, Table of n, a(n) for n = 1..1200

Prog

(PARI) forprime(p=2, 1000, if(A322906(p)==2, print1(p, ", ")))

Crossrefs

Cf. A006190, A175182, A322906, A322907.

Let {x(n)} be the sequence defined in the comment section.

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

Primes p such that w(p) = 1 | A112860* | A309580 | A309586

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

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: A296926 A063639 A230223 * A339954 A260893 A352630

Adjacent sequences: A309584 A309585 A309586 * A309588 A309589 A309590

Keyword

nonn

Author

Jianing Song, Aug 10 2019

Status

approved