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A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2*p+q)/5 is a prime and (r+2*s)/5^n is a prime.

%I A358198 #8 Jan 31 2023 08:26:14
%S A358198 11,101,243701,6758951,3257480201,5493848951,58634348951,218007942701,
%T A358198 21840280598951,213065296223951,186522444661451,383378987630201,
%U A358198 7794174397786451,110420241292317701,67327687581380201,91455128987630201,3987035878499348951,80659241994222005201,4289429982503255201
%N A358198 a(n) is the first member p of A007530 such that, with q = p+2, r = p+6 and s = p+8, (2*p+q)/5 is a prime and (r+2*s)/5^n is a prime.
%e A358198 a(3) = 243701 because p = 247301, q = p+2 = 247303, r = p+6 = 243707, s = p+8 = 243709, (2*p+q)/5 = 146221 and (r+2*s)/5^3 = 5849 are primes, and p is the least prime that works.
%p A358198 f:= proc(n) local t,p,m;
%p A358198         m:= 5^n;
%p A358198         t:= 3;
%p A358198         do
%p A358198           t:= nextprime(t);
%p A358198           if t*m mod 3 <> 1 then next fi;
%p A358198           p:= (t*m-22)/3;
%p A358198           if isprime(p) and isprime(p+2) and isprime(p+6) and isprime(p+8) and isprime((3*p+2)/5) then return p fi;
%p A358198         od;
%p A358198 end proc;
%p A358198 map(f, [$1..20]);
%t A358198 a[n_] := a[n] = Module[{t = 3, p, m = 5^n}, While[True, t = NextPrime[t]; If[Mod[t*m, 3] != 1, Continue[]]; p = (t*m - 22)/3; If[AllTrue[{p, p+2, p+6, p+8, (3p+2)/5}, PrimeQ], Return[p]]]];
%t A358198 Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* _Jean-François Alcover_, Jan 31 2023, after Maple program *)
%Y A358198 Cf. A007530, A358149.
%K A358198 nonn
%O A358198 1,1
%A A358198 _J. M. Bergot_ and _Robert Israel_, Nov 02 2022