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Integers k such that 19*(10^k) + 1 is prime.
1

%I #37 May 03 2024 07:04:26

%S 1,2,3,11,18,25,60,71,85,168,285,627,872,1092,1101,1169,1974,2011,

%T 2135,2405,10147,14029,16121,17501,46548

%N Integers k such that 19*(10^k) + 1 is prime.

%C For k > 0, numbers k such that the digits 19 followed by k-1 occurrences of the digit 0 followed by the digit 1 is prime (see Example section).

%C a(26) > 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 190w1</a>.

%e 3 is in this sequence because 19*10^3 + 1 = 19001 is prime.

%e Initial terms and associated primes:

%e a(1) = 1, 191;

%e a(2) = 2, 1901;

%e a(3) = 3, 19001;

%e a(4) = 11, 1900000000001;

%e a(5) = 18, 19000000000000000001, etc.

%p select(k->isprime(19*(10^k)+1),[$1..3000]); # _Muniru A Asiru_, Oct 07 2018

%t Select[Range[0, 100000], PrimeQ[19*10^# + 1] &]

%o (PARI) is(n) = ispseudoprime(19*(10^n) + 1); \\ _Altug Alkan_, Jan 15 2016

%o (PARI) lista(nn) = for(n=1, nn, if(isprime(19*(10^n) +1 ), print1(n, ", "))); \\ _Altug Alkan_, Jan 18 2016

%K nonn,more

%O 1,2

%A _Emre APARI_, Jan 14 2016

%E a(13)-a(20) from _Altug Alkan_, Jan 18 2016

%E a(21)-a(25) from _Robert Price_, Oct 07 2018