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A319450 Numbers k such that k and k + 1 both have primitive roots. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..59.

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.

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 A162228

Adjacent sequences:  A319447 A319448 A319449 * A319451 A319452 A319453

KEYWORD

nonn

AUTHOR

Jianing Song, Sep 19 2018

STATUS

approved

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Last modified January 20 11:11 EST 2020. Contains 331083 sequences. (Running on oeis4.)