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Numbers k such that (16*10^k + 167)/3 is prime.
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%I #11 Jun 02 2024 09:29:20

%S 0,1,5,8,12,13,14,15,33,98,123,260,485,1340,1674,1775,5988,7039,9421,

%T 15149,21751,30882,36517,85839,121633,131180,140091,188823

%N Numbers k such that (16*10^k + 167)/3 is prime.

%C For k > 1, numbers k such that the digit 5 followed by k-2 occurrences of the digit 3 followed by the digits 89 is prime (see Example section).

%C a(29) > 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 53w89</a>.

%e 5 is in this sequence because (16*10^5 + 167)/3 = 533389 is prime.

%e Initial terms and associated primes:

%e a(1) = 0, 61;

%e a(2) = 1, 109;

%e a(3) = 5, 533389;

%e a(4) = 8, 533333389;

%e a(5) = 12, 5333333333389; etc.

%t Select[Range[0, 100000], PrimeQ[(16*10^# + 167)/3] &]

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

%K nonn,more,hard

%O 1,3

%A _Robert Price_, Oct 25 2017

%E a(25)-a(28) from _Robert Price_, Jun 18 2019