A319450 Numbers k such that k and k + 1 both have primitive roots.

1, 2, 3, 4, 5, 6, 9, 10, 13, 17, 18, 22, 25, 26, 37, 46, 49, 53, 58, 61, 73, 81, 82, 97, 106, 121, 157, 162, 166, 178, 193, 226, 241, 242, 250, 262, 277, 313, 337, 346, 358, 361, 382, 397, 421, 457, 466, 478, 486, 502, 541, 562, 577, 586, 613, 625, 661, 673, 718

Offset: 1

Author

Jianing Song, Sep 19 2018

Keywords

• nonn

Comments

Numbers k such that both k and k + 1 are in A033948.

Apart from the first four terms, numbers k such that there exists odd primes p, q and positive numbers u, v such that k = p^u, k + 1 = 2*q^v or k = 2*p^u, k + 1 = q^v.

Let p be an odd prime. If 2*p^e - 1 is prime, then 2*p^e - 1 is a term. If 2*p^e + 1 is prime, then 2*p^e is a term. If (p^2^e + 1)/2 is prime, then p^2^e is a term. However it's not known whether there are infinitely many primes of the form 2*p^e +- 1 or (p^2^e + 1)/2.

The case that k and k + 1 are both in this sequence is extremely rare. Only 11 such k are known: 1, 2, 3, 4, 5, 9, 17, 25, 81, 241 and 3^541 - 2. It's possible that there are no further members. See A305237.

Examples

			5 is a primitive root modulo both 46 and 47, so 46 is a term.
2 is a primitive root modulo 53 and 5 is a primitive root modulo 54, so 53 is a term.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1001 from Michel Marcus)

Crossrefs

Cf. A033948, A305237.

Programs

Mathematica

q[n_] := q[n] = EulerPhi[n] == CarmichaelLambda[n]; Select[Range[720],q[#] && q[# + 1] &] (* Amiram Eldar, Jul 21 2024 *)

PARI

isA033948(n) = (#znstar(n)[2]<=1)
isA319450(n) = isA033948(n)&&isA033948(n+1)