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

%I A361914 #20 Apr 23 2023 23:44:09
%S A361914 7,13,43,73,127,157,211,241,307,421,463,601,757,1093,1123,1483,1723,
%T A361914 2551,2801,2971,3307,3541,3907,4423,4831,5113,5701,6007,6163,6481,
%U A361914 8011,9901,10303,11131,12211,12433,13807,14281,17293,19183,19531,20023,20593,21757,22621,22651,23563
%N A361914 Primes that are repunits with three or more digits for exactly one base b >= 2.
%C A361914 Brazilian primes that have exactly one Brazilian representation as a repunit.
%C A361914 As these primes p satisfy beta(p) = tau(p) / 2 (= 1), where beta = A220136 and tau = A000005, this sequence is a subsequence of A326380.
%C A361914 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.
%C A361914 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).
%H A361914 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goormaghtigh_conjecture">Goormaghtigh conjecture</a>.
%H A361914 <a href="/index/Br#Brazilian_numbers">Index entries for sequences related to Brazilian numbers</a>.
%e A361914 7 = 111_2 is a term.
%e A361914 13 = 111_3 is a term.
%e A361914 19 = 11_18 is not a term.
%e A361914 31 = 11111_5 = 111_5 is not a term.
%e A361914 127 = 1111111_2 is a term.
%e A361914 8191 = 1111111111111_2 = 111_90 is not a term.
%t A361914 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 *)
%Y A361914 Cf. A000005, A085104, A220136, A220627, A326380.
%Y A361914 Equals A326380 \ {A326385 Union A326387}.
%Y A361914 Subsequence of A288783.
%K A361914 nonn,base,less
%O A361914 1,1
%A A361914 _Bernard Schott_, Mar 29 2023