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Numbers k such that 4*10^k - 99 is prime.
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%I #20 Jun 07 2024 08:44:22

%S 4,6,7,9,43,53,67,69,123,178,201,448,649,658,1068,2003,2584,4445,4816,

%T 6322,8477,8868,14500,24934,33412,42504,87287,87337

%N Numbers k such that 4*10^k - 99 is prime.

%C For k > 1, numbers k such that the digit 3 followed by k-2 occurrences of the digit 9 followed by the digits 01 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 39w01</a>.

%e 4 is in this sequence because 4*10^4 - 99 = 39901 is prime.

%e Initial terms and associated primes:

%e a(1) = 4, 39901;

%e a(2) = 6, 3999901;

%e a(3) = 7, 39999901;

%e a(4) = 9, 3999999901;

%e a(5) = 43, 39999999999999999999999999999999999999999901; etc.

%t Select[Range[0, 100000], PrimeQ[4*10^# - 99] &]

%o (Magma) [n: n in [2..500] | IsPrime(4*10^n - 99)]; // _Vincenzo Librandi_, Dec 03 2016

%o (PARI) is(n)=ispseudoprime(4*10^n - 99) \\ _Charles R Greathouse IV_, Jun 13 2017

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

%K nonn,more,hard

%O 1,1

%A _Robert Price_, Dec 02 2016

%E Link and Mathematica program corrected by _Robert Price_, Jun 11 2018