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A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p. 10
5, 13, 29, 37, 41, 73, 89, 97, 109, 137, 149, 157, 181, 193, 197, 229, 233, 241, 269, 281, 293, 317, 349, 353, 373, 389, 397, 401, 409, 421, 449, 457, 461, 509, 541, 557, 577, 593, 613, 617, 653, 661, 701, 709, 733, 761, 769, 773, 797, 821, 853, 857, 877 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Primes p such that A322906(p) = 4.

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

(a) A175182(p) == 4 (mod 8);

(b) ord(p,(3+sqrt(13))/2) == 4 (mod 8), 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) == 2 (mod 4);

(d) A322907(p) is odd;

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

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) = 4;

(2) pi(p) == 4 (mod 8);

(3) ord(p,u) == 4 (mod 8);

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

(5) r(p) is odd;

(6) ord(p,-u^2) is odd.

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 5, 21, 33, 37, 41, 45 modulo 52.

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

LINKS

Table of n, a(n) for n=1..53.

PROG

(PARI) forprime(p=2, 900, if(A322906(p)==4, 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  | A309587

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

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: A224339 A133204 A207040 * A268614 A152658 A100877

Adjacent sequences:  A309585 A309586 A309587 * A309589 A309590 A309591

KEYWORD

nonn

AUTHOR

Jianing Song, Aug 10 2019

STATUS

approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)