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A309586 Primes p with 1 zero in a fundamental period of A006190 mod p. 12
2, 3, 23, 43, 53, 61, 79, 101, 103, 107, 127, 131, 139, 173, 179, 191, 199, 211, 251, 263, 277, 283, 311, 347, 367, 419, 433, 439, 443, 467, 491, 503, 523, 547, 563, 569, 571, 599, 607, 647, 659, 677, 719, 727, 751, 757, 823, 829, 859, 881, 883, 887, 907 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that A322906(p) = 1.

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

(a) A175182(p) == 2 (mod 4);

(b) ord(p,(3+sqrt(13))/2) == 2 (mod 4), where ord(p,u) is the smallest integer k > 0 such that (u^k - 1)/p is an algebraic integer;

(c) ord(p,(11+3*sqrt(13))/2) is odd;

(d) A322907(p) == 2 (mod 4);

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

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) = 1;

(2) pi(p) == 2 (mod 4);

(3) ord(p,u) == 2 (mod 4);

(4) ord(p,u^2) is odd;

(5) r(p) == 2 (mod 4);

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

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, 23, 27, 35, 43, 51 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, 900, if(A322906(p)==1, 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  | this seq

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: A215353 A215305 A215282 * A090708 A199342 A262838

Adjacent sequences:  A309583 A309584 A309585 * A309587 A309588 A309589

KEYWORD

nonn

AUTHOR

Jianing Song, Aug 10 2019

STATUS

approved

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Last modified September 19 06:21 EDT 2021. Contains 347551 sequences. (Running on oeis4.)