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Numbers k such that 22*10^k + 3 is prime.
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%I #14 May 25 2024 17:39:09

%S 1,2,3,6,15,18,26,51,54,68,85,242,597,2335,3463,4160,5565,8322,11389,

%T 43982,70729,127645,151998

%N Numbers k such that 22*10^k + 3 is prime.

%C For k > 0, numbers k such that the digits 22 followed by k-1 occurrences of the digit 0 followed by the digit 3 is prime (see Example section).

%C a(24) > 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 220w3</a>.

%e 2 is in this sequence because 22*10^2 + 3 = 2203 is prime.

%e Initial terms and associated primes:

%e a(1) = 1, 223;

%e a(2) = 2, 2203;

%e a(3) = 3, 22003;

%e a(4) = 6, 22000003;

%e a(5) = 15, 22000000000000003; etc.

%t Select[Range[0, 100000], PrimeQ[22*10^# + 3] &]

%o (PARI) isok(k) = isprime(22*10^k + 3); \\ _Altug Alkan_, Oct 11 2017

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

%K nonn,more,hard

%O 1,2

%A _Robert Price_, Oct 11 2017

%E a(22)-a(23) from _Robert Price_, Jan 04 2020