%I #43 Jun 30 2024 16:12:21
%S 0,1,2,3,4,11,14,16,92,133,153,378,448,785,1488,1915,2297,3286,4755,
%T 5825,7820,34442,34941
%N Numbers k such that 1 + 3*10^k + 100^k is prime.
%C All primes were certified with WinPFGW.
%C a(24) > 35000. - _Serge Batalov_, Dec 20 2015
%H Chris K. Caldwell, <a href="https://primes.utm.edu/primes/search.php?Description=%5E10%5E%25%2B3*10%5E%25%2B1&OnList=all&Number=20&Style=HTML">Prime Pages, Database Search Output</a>
%e 4 is in the sequence because 10^8 + 3 * 10^4 + 1 = 100030001 is prime.
%t Select[Range@ 1000, PrimeQ[1 + 3 10^# + 100^#] &] (* _Michael De Vlieger_, Dec 18 2015 *)
%o (PARI) \\sieve for the candidates:
%o {
%o lim=10^9; ns=6*10^5; pp=10^7; s=vectorsmall(ns);
%o forprime(p=11,lim,if(kronecker(5,p)==1,o=znorder(t=Mod(10,p));
%o q=sqrt(Mod(5,p));r=znlog((q-3)/2,t,o);
%o if(r,forstep(n=r,ns,o,s[n]=1);forstep(n=o-r,ns,o,s[n]=1)));
%o if(p>pp,pp+=10000000;print1(p" ")));
%o for(n=1,ns,if(!s[n],write("sieve_out_10301NGm1.txt", n)));
%o }
%o \\quick initial check for small sequence members
%o for(n=0,2297,if(ispseudoprime((10^n+3)*10^n+1),print1(n", ")))
%o \\ _Serge Batalov_, Dec 17 2015
%o (Magma) [n: n in [0..4*10^2] | IsPrime(1+3*10^n+100^n)]; // _Vincenzo Librandi_, Dec 22 2015
%Y Cf. A082622.
%Y Cf. A171411, A171459, A171514, A171554.
%K more,nonn
%O 1,3
%A _Jason Earls_, Dec 07 2009
%E a(21)-a(23) from _Serge Batalov_, Dec 20 2015