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Numbers k such that (10^k - 13)/3 is prime.
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%I #30 May 25 2024 17:37:05

%S 2,4,5,8,11,25,35,270,401,613,635,768,1283,2941,3409,4266,4391,10744,

%T 22979,26766,27743,35514,59174,86906,99239,154494

%N Numbers k such that (10^k - 13)/3 is prime.

%C For k > 1, numbers k such that k-2 occurrences of the digit 3 followed by the digits 29 is prime (see Example section).

%C a(27) > 2 * 10^5.

%C There are no odd multiples of 3 in this sequence. If k is an odd multiple of 3, then (10^k - 13)/3 is divisible by 7. The smallest even multiple of 3 in the sequence is 270. - _Alonso del Arte_, Dec 31 2017

%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 3w29</a>.

%e 4 is in this sequence because (10^4 - 13)/3 = 3329 is prime.

%e Here is a listing of the initial terms and associated primes:

%e a(1) = 2, 29;

%e a(2) = 4, 3329;

%e a(3) = 5, 33329;

%e a(4) = 8, 33333329;

%e a(5) = 11, 33333333329; etc.

%t Select[Range[2, 100000], PrimeQ[(10^# - 13)/3] &]

%o (PARI) isok(n) = isprime((10^n-13)/3); \\ _Altug Alkan_, Dec 31 2017

%Y Cf. A056707 (numerators of (10^k - 13)/3 that are prime for k negative), A056654, A268448, A269303, A270339, A270613, A270831, A270890, A270929, A271269.

%K nonn,more,hard

%O 1,1

%A _Robert Price_, Feb 21 2017

%E a(26) from _Robert Price_, Dec 31 2017