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A309587
Primes p with 2 zeros in a fundamental period of A006190 mod p.
19
7, 11, 17, 19, 31, 47, 59, 67, 71, 83, 113, 151, 163, 167, 223, 227, 239, 257, 271, 307, 313, 331, 337, 359, 379, 383, 431, 463, 479, 487, 499, 521, 587, 601, 619, 631, 641, 643, 673, 683, 691, 739, 743, 787, 809, 811, 827, 839, 863, 947, 967, 983
OFFSET
1,1
COMMENTS
Primes p such that A322906(p) = 2.
For p > 2, p is in this sequence if and only if 8 divides A175182(p), and if and only if 4 divides A322907(p). For a proof of the equivalence between A322906(p) = 2 and 4 dividing A322907(p), see Section 2 of my link below.
This sequence contains all primes congruent to 7, 11, 15, 19, 31, 47 modulo 52. This corresponds to case (2) for k = 11 in the Conclusion of Section 1 of my link below.
Conjecturely, this sequence has density 1/3 in the primes. [Comment rewritten by Jianing Song, Jun 16 2024 and Jun 25 2024]
PROG
(PARI) forprime(p=2, 1000, if(A322906(p)==2, print1(p, ", ")))
CROSSREFS
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 | this seq
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: A296926 A063639 A230223 * A339954 A260893 A352630
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 10 2019
STATUS
approved