A309580 Primes p with 1 zero in a fundamental period of A000129 mod p.

2, 7, 23, 31, 41, 47, 71, 79, 103, 127, 151, 167, 191, 199, 223, 239, 263, 271, 311, 313, 353, 359, 367, 383, 409, 431, 439, 457, 463, 479, 487, 503, 599, 607, 631, 647, 719, 727, 743, 751, 761, 809, 823, 839, 863, 887, 911, 919, 967, 983, 991, 1031, 1039, 1063, 1087, 1103, 1129, 1151, 1201, 1223, 1231, 1279

Offset

1,1

Comments

Primes p such that A214027(p) = 1.

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

(a) A175181(p) == 2 (mod 4);

(b) ord(p,1+sqrt(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,3+2*sqrt(2)) is odd;

(d) A214028(p) == 2 (mod 4);

(e) ord(p,-(3+2*sqrt(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 7 modulo 8.

Conjecturely, this sequence has density 7/24 in the primes.

Links

Jianing Song, Table of n, a(n) for n = 1..1280

Prog

(PARI) forprime(p=2, 1300, if(A214027(p)==1, 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* | this seq | A309586

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: A045381 A180537 A042145 * A186098 A040098 A045315

Adjacent sequences: A309577 A309578 A309579 * A309581 A309582 A309583

Keyword

nonn

Author

Jianing Song, Aug 10 2019

Status

approved