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 A309588 Primes p with 4 zeros in a fundamental period of A006190 mod p. 10
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Primes p such that A322906(p) = 4. For p > 2, p is in this sequence if and only if (all these conditions are equivalent): (a) A175182(p) == 4 (mod 8); (b) ord(p,(3+sqrt(13))/2) == 4 (mod 8), where ord(p,u) is the smallest integer k > 0 such that (u^k - 1)/p is an algebraic integer; (c) ord(p,(11+3*sqrt(13))/2) == 2 (mod 4); (d) A322907(p) is odd; (e) ord(p,-(11+3*sqrt(13))/2) is odd. In general, let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = m*x(n+1) + x(n). Let pi(k) be the Pisano period of {x(n)} modulo k, i.e., pi(k) = min{l > 0 : x(n+l) == x(n) (mod k) for all n}, r(k) = min{l > 0 : k divides x(l)} and w(k) be the number of zeros in a fundamental period of {x(n)} modulo k. Let u = (m + sqrt(m^2+4))/2, p be an odd prime, then these conditions are equivalent: (1) w(p) = 4; (2) pi(p) == 4 (mod 8); (3) ord(p,u) == 4 (mod 8); (4) ord(p,u^2) == 2 (mod 4); (5) r(p) is odd; (6) ord(p,-u^2) is odd. This can be shown by noting that pi(p) = p^c*ord(p,u) and r(p) = p^c*ord(p,-u^2) for some c (if p does not divide m^2 + 4 then c = 0, otherwise c = 1). Also, Pi(p) is always even, so ord(p,u) is always even. This sequence contains all primes congruent to 5, 21, 33, 37, 41, 45 modulo 52. Conjecturely, this sequence has density 1/3 in the primes. LINKS PROG (PARI) forprime(p=2, 900, if(A322906(p)==4, print1(p, ", "))) CROSSREFS Cf. A006190, A175182, A322906, A322907. Let {x(n)} be the sequence defined in the comment section.                              |   m=1    |   m=2    |   m=3 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: A224339 A133204 A207040 * A268614 A152658 A100877 Adjacent sequences:  A309585 A309586 A309587 * A309589 A309590 A309591 KEYWORD nonn AUTHOR Jianing Song, Aug 10 2019 STATUS approved

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Last modified January 29 16:58 EST 2020. Contains 331347 sequences. (Running on oeis4.)