A361914 Primes that are repunits with three or more digits for exactly one base b >= 2.

7, 13, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023, 20593, 21757, 22621, 22651, 23563

Offset: 1

Author

Bernard Schott, Mar 29 2023

Keywords

• nonn
• base
• less

Comments

Brazilian primes that have exactly one Brazilian representation as a repunit.

As these primes p satisfy beta(p) = tau(p) / 2 (= 1), where beta = A220136 and tau = A000005, this sequence is a subsequence of A326380.

Equals A085104 \ {31, 8191}, since according to the Goormaghtigh conjecture (link), 31 and 8191 which are both Mersenne numbers, are the only primes which are Brazilian in two different bases.

The three following sequences realize a partition of the set of primes: A220627 (primes not Brazilian), this sequence (primes 1-Brazilian) and {31,8191} (primes 2-Brazilian).

Examples

			7 = 111_2 is a term.
13 = 111_3 is a term.
19 = 11_18 is not a term.
31 = 11111_5 = 111_5 is not a term.
127 = 1111111_2 is a term.
8191 = 1111111111111_2 = 111_90 is not a term.
		

Links

• Wikipedia, Goormaghtigh conjecture.
• Index entries for sequences related to Brazilian numbers.

Crossrefs

Cf. A000005, A085104, A220136, A220627, A326380.

Equals A326380 \ {A326385 Union A326387}.

Subsequence of A288783.

Programs

Mathematica

q[n_] := Count[Range[2, n - 2], ?(Length[d = IntegerDigits[n, #]] > 2 && Length[Union[d]] == 1 &)] == 1;; Select[Prime[Range[3000]], q] (* _Amiram Eldar, Mar 30 2023 *)