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Numbers k such that (29*10^k + 67)/3 is prime.
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%I #27 Jun 10 2024 23:36:06

%S 3,6,8,10,16,80,105,123,130,159,160,300,597,1653,3576,4535,6164,8601,

%T 24203,24636,165796,171275

%N Numbers k such that (29*10^k + 67)/3 is prime.

%C For k > 1, numbers k such that the digit 9 followed by k-2 occurrences of the digit 6 followed by the digits 89 is prime (see Example section).

%C a(23) > 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 96w89</a>.

%e 3 is in this sequence because (29*10^3 + 67)/3 = 9689 is prime.

%e Initial terms and associated primes:

%e a(1) = 3, 9689;

%e a(2) = 6, 9666689;

%e a(3) = 8, 966666689;

%e a(4) = 10, 96666666689;

%e a(5) = 16, 96666666666666689; etc.

%t Select[Range[0, 100000], PrimeQ[(29*10^# + 67)/3] &]

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

%K nonn,more,hard

%O 1,1

%A _Robert Price_, Oct 09 2017

%E a(21)-a(22) from _Robert Price_, Dec 18 2019