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A309581 Primes p with 2 zeros in a fundamental period of A000129 mod p. 12
3, 11, 17, 19, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 331, 337, 347, 379, 401, 419, 433, 443, 449, 467, 491, 499, 523, 547, 563, 571, 577, 587, 601, 617, 619, 641, 643, 659, 673, 683, 691 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that A214027(p) = 2.

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

(a) 8 divides A175181(p);

(b) 8 divides ord(p,1+sqrt(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,3+2*sqrt(2));

(d) 4 divides A214028(p);

(e) 4 divides ord(p,-(3+2*sqrt(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 3 modulo 8.

Conjecturely, this sequence has density 5/12 in the primes.

LINKS

Jianing Song, Table of n, a(n) for n = 1..1600

PROG

(PARI) forprime(p=2, 700, if(A214027(p)==2, print1(p, ", ")))

CROSSREFS

Cf. A000129, A175181, A214027, A214028.

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  | this seq | 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: A085317 A210311 A033200 * A291277 A191375 A260793

Adjacent sequences:  A309578 A309579 A309580 * A309582 A309583 A309584

KEYWORD

nonn

AUTHOR

Jianing Song, Aug 10 2019

STATUS

approved

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Last modified October 23 13:32 EDT 2021. Contains 348214 sequences. (Running on oeis4.)