cp's OEIS Frontend

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

%I A261580 #64 Jun 27 2025 21:00:12
%S A261580 5,13,29,37,53,61,101,109,137,149,157,173,181,197,229,269,277,293,317,
%T A261580 349,373,389,397,421,461,509,521,541,557,569,593,613,653,661,677,701,
%U A261580 709,733,757,773,797,821,829,853,857,877,941,953,997,1013,1021,1061,1069
%N A261580 Primes p such that A214028(p) is odd.
%C A261580 From _Jianing Song_, Aug 13 2019: (Start)
%C A261580 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.
%C A261580 For p > 2, p is in this sequence if and only if A175181(p) == 4 (mod 8).
%C A261580 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.
%C A261580 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]
%C A261580 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
%H A261580 Jianing Song, <a href="/A261580/b261580.txt">Table of n, a(n) for n = 1..1280</a>
%H A261580 Bernadette Faye and Florian Luca, <a href="http://arxiv.org/abs/1508.05714">Pell Numbers whose Euler Function is a Pell Number</a>, arXiv:1508.05714 [math.NT], 2015.
%H A261580 Jianing Song, <a href="/A053027/a053027.pdf">Lucas sequences and entry point modulo p</a>
%e A261580 The smallest Pell number divisible by the prime 5 has index 3, which is odd, so 5 is in the sequence.
%t A261580 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 *)
%o A261580 (PARI) pell(n) = polcoeff(Vec(x/(1-2*x-x^2) + O(x^(n+1))), n);
%o A261580 z(n) = {k=1; while (pell(k) % n, k++); k;}
%o A261580 lista(nn) = {forprime(p=2, nn, if (z(p) % 2, print1(p, ", ")););}
%o A261580 (PARI) forprime(p=2, 1100, if(A214027(p)==4, print1(p, ", "))) \\ _Jianing Song_, Aug 13 2019
%Y A261580 Cf. A214028, A261581.
%Y A261580 Cf. also A175181.
%Y A261580 Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let w(k) be the number of zeros in a fundamental period of {x(n)} modulo k.
%Y A261580                              |   m=1    |   m=2    |   m=3
%Y A261580 -----------------------------+----------+----------+---------
%Y A261580 The sequence {x(n)}          | A000045  | A000129  | A006190
%Y A261580 The sequence {w(k)}          | A001176  | A214027  | A322906
%Y A261580 Primes p such that w(p) = 1  | A112860* | A309580  | A309586
%Y A261580 Primes p such that w(p) = 2  | A053027  | A309581  | A309587
%Y A261580 Primes p such that w(p) = 4  | A053028  | this seq | A309588
%Y A261580 Numbers k such that w(k) = 1 | A053031  | A309583  | A309591
%Y A261580 Numbers k such that w(k) = 2 | A053030  | A309584  | A309592
%Y A261580 Numbers k such that w(k) = 4 | A053029  | A309585  | A309593
%Y A261580 * and also A053032 U {2}
%K A261580 nonn
%O A261580 1,1
%A A261580 _Michel Marcus_, Aug 25 2015