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A309588
Primes p with 4 zeros in a fundamental period of A006190 mod p.
19
5, 13, 29, 37, 41, 73, 89, 97, 109, 137, 149, 157, 181, 193, 197, 229, 233, 241, 269, 281, 293, 317, 349, 353, 373, 389, 397, 401, 409, 421, 449, 457, 461, 509, 541, 557, 577, 593, 613, 617, 653, 661, 701, 709, 733, 761, 769, 773, 797, 821, 853, 857, 877
OFFSET
1,1
COMMENTS
Primes p such that A322906(p) = 4.
For p > 2, p is in this sequence if and only if A175182(p) == 4 (mod 8), and if and only if A322907(p) is odd. For a proof of the equivalence between A322906(p) = 4 and A322907(p) being odd, see Section 2 of my link below.
This sequence contains all primes congruent to 5, 21, 33, 37, 41, 45 modulo 52. This corresponds to case (1) 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, 900, if(A322906(p)==4, 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 | A309587
Primes p such that w(p) = 4 | A053028 | A261580 | this seq
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: A368546 A133204 A207040 * A268614 A152658 A347836
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 10 2019
STATUS
approved