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Numbers k such that 9*10^k + 89 is prime.
0

%I #15 May 25 2024 14:35:08

%S 1,4,5,7,9,15,26,43,75,106,129,141,147,171,351,512,927,1156,2563,2689,

%T 3274,4211,5709,6908,11781,14464,31451,34921,36635,38087,55707,59617,

%U 104361,123549

%N Numbers k such that 9*10^k + 89 is prime.

%C For k > 1, numbers k such that the digit 9 followed by k-2 occurrences of the digit 0 followed by the digits 89 is prime (see Example section).

%C a(35) > 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 90w89</a>.

%e 4 is in this sequence because 9*10^4 + 89 = 90089 is prime.

%e Initial terms and associated primes:

%e a(1) = 1, 179;

%e a(2) = 4, 90089;

%e a(3) = 5, 900089;

%e a(4) = 7, 90000089;

%e a(5) = 9, 9000000089; etc.

%t Select[Range[0, 100000], PrimeQ[9*10^# + 89] &]

%o (PARI) is(n)=ispseudoprime(9*10^n + 89) \\ _Charles R Greathouse IV_, Jun 13 2017

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

%K nonn,more,hard

%O 1,2

%A _Robert Price_, Nov 18 2016

%E a(33)-a(34) from _Robert Price_, Nov 09 2019