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A261580
Primes such that z(p) is odd where z(n) is A214028(n).
20
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, this sequence has density 7/24 in the primes. (End) [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 20 2024]
LINKS
Bernadette Faye and Florian Luca, Pell Numbers whose Euler Function is a Pell Number, arXiv:1508.05714 [math.NT], 2015.
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. also A175181.
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.
| m=1 | m=2 | m=3
-----------------------------+----------+----------+---------
The sequence {x(n)} | A000045 | A000129 | A006190
The sequence {w(k)} | A001176 | A214027 | A322906
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 | this seq | 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: A152658 A347836 A100877 * A007521 A294919 A213050
KEYWORD
nonn
AUTHOR
Michel Marcus, Aug 25 2015
STATUS
approved