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A019335
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Primes with primitive root 5.
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10
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2, 3, 7, 17, 23, 37, 43, 47, 53, 73, 83, 97, 103, 107, 113, 137, 157, 167, 173, 193, 197, 223, 227, 233, 257, 263, 277, 283, 293, 307, 317, 347, 353, 373, 383, 397, 433, 443, 463, 467, 503, 523, 547, 557, 563, 577, 587, 593, 607, 613, 617, 647, 653, 673, 677, 683, 727
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OFFSET
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1,1
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COMMENTS
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To allow primes less than the specified primitive root m (here, 5) to be included, we use the essentially equivalent definition "Primes p such that the multiplicative order of m mod p is p-1". This comment applies to all of A019334-A019421. - N. J. A. Sloane, Dec 02 2019
Appears to be the numbers k such that the sequence 5^n mod k has period length k-1. All terms are congruent to 2 or 3 mod 5. - Gary Detlefs, May 21 2014
If we define
Pi(N,b) = # {p prime, p <= N, p == b (mod 5)};
Q(N) = # {p prime, p <= N, p in this sequence},
then by Artin's conjecture, Q(N) ~ (20/19)*C*N/log(N) ~ (40/19)*C*(Pi(N,2) + Pi(N,3)), where C = A005596 is Artin's constant.
Conjecture: if we further define
Q(N,b) = # {p prime, p <= N, p == b (mod 5), p in this sequence},
then we have:
Q(N,2) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,2);
Q(N,3) ~ (1/2)*Q(N) ~ (20/19)*C*Pi(N,3). (End)
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LINKS
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MATHEMATICA
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pr=5; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == #-1 &]
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PROG
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(PARI) isok(p) = isprime(p) && (p != 5) && (znorder(Mod(5, p)) == p-1); \\ Michel Marcus, Apr 27 2019
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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