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Numbers k such that (19*10^k - 37)/9 is prime.
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%I #16 Jun 08 2024 05:45:07

%S 1,4,6,10,16,37,60,64,78,96,166,256,1294,1398,2044,2244,5080,7464,

%T 8041,17929,18144,29080,32623

%N Numbers k such that (19*10^k - 37)/9 is prime.

%C For k > 1, numbers k such that the digit 2 followed by k-2 occurrences of the digit 1 followed by the digits 07 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 21w07</a>.

%e 4 is in this sequence because (19*10^4 - 37)/9 = 21107 is prime.

%e Initial terms and associated primes:

%e a(1) = 1, 17;

%e a(2) = 4, 21107;

%e a(3) = 6, 2111107;

%e a(4) = 10, 21111111107;

%e a(5) = 16, 21111111111111107; etc.

%t Select[Range[1, 100000], PrimeQ[(19*10^# - 37)/9] &]

%o (PARI) isok(n) = isprime((19*10^n - 37)/9); \\ _Indranil Ghosh_, Mar 09 2017

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

%K nonn,more,hard

%O 1,2

%A _Robert Price_, Mar 08 2017