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Numbers k such that (10^k - 19)/9 is prime.
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%I #20 May 26 2024 22:59:10

%S 3,4,6,9,34,36,126,141,160,286,678,1236,3757,4681,4881,6360,6588,7056,

%T 7752,41925,43423,43503,47565,52608,55084,76069,199248

%N Numbers k such that (10^k - 19)/9 is prime.

%C For k > 1, numbers k such that k-2 occurrences of the digit 1 followed by the digits 09 is prime (see Example section).

%C a(28) > 2*10^5.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr">Factorization of near-repdigit-related numbers</a>.

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/prime/prime_difficulty.txt">Search for 1w09</a>.

%e 4 is in this sequence because (10^4 - 19)/9 = 1109 is prime.

%e Initial terms and associated primes:

%e a(1) = 3, 109;

%e a(2) = 4, 1109;

%e a(3) = 6, 111109;

%e a(4) = 9, 111111109;

%e a(5) = 34, 1111111111111111111111111111111109; etc.

%t Select[Range[2, 100000], PrimeQ[10^# - 19)/9] &]

%Y Cf. A056654, A268448, A269303, A270339, A270613, A270831, A270890, A270929, A271269.

%K nonn,more,hard

%O 1,1

%A _Robert Price_, Oct 06 2017

%E a(27) from _Robert Price_, Jan 02 2018