

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
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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



