A243755 Primes p such that p is a primitive root modulo the next prime p' and also p' is a primitive root modulo p.

2, 3, 5, 11, 59, 61, 83, 101, 131, 151, 179, 181, 197, 251, 257, 269, 271, 317, 337, 347, 367, 419, 443, 461, 523, 563, 577, 587, 593, 659, 709, 733, 797, 811, 821, 827, 863, 947, 971, 977, 1061, 1063, 1069, 1097, 1129, 1153, 1171, 1187, 1217, 1229, 1277, 1283, 1301, 1361, 1433, 1451, 1543, 1553, 1601, 1619

Offset

1,1

Comments

Conjecture: The sequence contains infinitely many primes. Moreover, there are infinitely many primes p such that both p and -p are primitive roots modulo the next prime p' and both p' and -p' are primitive roots modulo p.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, New observations on primitive roots modulo primes, arXiv:1405.0290 [math.NT], 2014.

Example

a(1) = 2 since prime(1) = 2 is a primitive root modulo prime(2) = 3 and also prime(2) = 3 is a primitive root modulo prime(1) = 2.
a(2) = 3 since prime(2) = 3 is a primitive root modulo prime(3) = 5 and also prime(3) = 5 is a primitive root modulo prime(2) = 3.

Mathematica

dv[n_]:=Divisors[n]
n=0; Do[Do[If[Mod[(Prime[m])^(Part[dv[Prime[m+1]-1], i]), Prime[m+1]]==1, Goto[aa]], {i, 1, Length[dv[Prime[m+1]-1]]-1}]; Do[If[Mod[Prime[m+1]^(Part[dv[Prime[m]-1], j]), Prime[m]]==1, Goto[aa]], {j, 1, Length[dv[Prime[m]-1]]-1}]; n=n+1; Print[n, " ", Prime[m]]; Label[aa]; Continue, {m, 1, 256}]

Crossrefs

Cf. A000040, A243164, A243403.

Sequence in context: A065296 A114895 A083685 * A242334 A087524 A088053

Adjacent sequences: A243752 A243753 A243754 * A243756 A243757 A243758

Keyword

nonn

Author

Zhi-Wei Sun, Jun 09 2014

Status

approved