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

1,2

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.

Links

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Cf. A033948, A305237.

Sequence in context: A178971 A082642 A217348 * A153013 A052492 A339454

Adjacent sequences: A319447 A319448 A319449 * A319451 A319452 A319453

Keyword

nonn

Author

Jianing Song, Sep 19 2018

Status

approved