

A361914


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


0



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
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OFFSET

1,1


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 1Brazilian) and {31,8191} (primes 2Brazilian).


LINKS



EXAMPLE

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.


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 *)


CROSSREFS



KEYWORD

nonn,base,less


AUTHOR



STATUS

approved



