A261580 Primes p such that A214028(p) is odd.

5, 13, 29, 37, 53, 61, 101, 109, 137, 149, 157, 173, 181, 197, 229, 269, 277, 293, 317, 349, 373, 389, 397, 421, 461, 509, 521, 541, 557, 569, 593, 613, 653, 661, 677, 701, 709, 733, 757, 773, 797, 821, 829, 853, 857, 877, 941, 953, 997, 1013, 1021, 1061, 1069

Offset

1,1

Comments

From Jianing Song, Aug 13 2019: (Start)

Primes p with 4 zeros in a fundamental period of A000129 mod p, that is, primes p such that A214027(p) = 4. For a proof of the equivalence between A214027(p) = 4 and A214028(p) being odd, see Section 2 of my link below.

For p > 2, p is in this sequence if and only if A175181(p) == 4 (mod 8).

This sequence contains all primes congruent to 5 modulo 8. This corresponds to case (1) for k = 6 in the Conclusion of Section 1 of my link below.

Conjecturely, since (k+2)/2 = 4 is a square, this sequence has density 7/24 in the primes; see the end of Section 1 of my link. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 20 2024]

The conjecture above is an analog of Hasse's result that the set {p prime : ord(2,p) is odd} has density 7/24 in the primes, where ord(a,m) is the multiplicative order of a modulo m; see A014663. - Jianing Song, Jun 26 2025

Links

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

Bernadette Faye and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.

Jianing Song, Entry points and periods of Lucas sequences

Example

The smallest Pell number divisible by the prime 5 has index 3, which is odd, so 5 is in the sequence.

Mathematica

f[n_] := Block[{k = 1}, While[Mod[Simplify[((1 + Sqrt@ 2)^k - (1 - Sqrt@ 2)^k)/(2 Sqrt@ 2)], n] != 0, k++]; k]; Select[Prime@ Range@ 180, OddQ@ f@ # &] (* Michael De Vlieger, Aug 25 2015 *)

Prog

(PARI) pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
z(n) = {k=1; while (pell(k) % n, k++); k; }
lista(nn) = {forprime(p=2, nn, if (z(p) % 2, print1(p, ", ")); ); }
(PARI) forprime(p=2, 1100, if(A214027(p)==4, print1(p, ", "))) \\ Jianing Song, Aug 13 2019

Crossrefs

Cf. A214028, A261581.

Cf. also A175181.

For a list of sequences related to the numbers of zeros in a fundamental period of {x(n)}, where {x(n)} is a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n), see A053032.

Sequence in context: A152658 A347836 A100877 * A007521 A294919 A213050

Adjacent sequences: A261577 A261578 A261579 * A261581 A261582 A261583

Keyword

nonn,changed

Author

Michel Marcus, Aug 25 2015

Status

approved