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a(n) is the first prime p such that, if q is the next prime, (p*q+p+q)/5^n is a prime.
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%I #63 Dec 30 2022 06:32:40

%S 2,7,1847,90793,139313,1790293,3834043,5521543,24996487,2062865293,

%T 5555052793,12111965183,95460776921,6045070151921,10204150316653,

%U 70501997496487,442748358250633,368313674465183,2935956099058987,10360552690003447,120999670013476223,1820610211470152737

%N a(n) is the first prime p such that, if q is the next prime, (p*q+p+q)/5^n is a prime.

%C Suggested in an email by _J. M. Bergot_.

%C For n >= 1, a(n) has the form k * 5^n + x, for some k >= 0, where x is a solution to the modular quadratic equation x^2 + (d+2)*x + d == 0 (mod 5^n), where d = q-p. - _Daniel Suteu_, Dec 28 2022

%e a(2) = 1847 because 1847 is prime, the next prime is 1861, 1847*1861 + 1847 + 1861 = 3440975 = 5^2 * 137639 where 137639 is prime, and no smaller prime works.

%p V:= Array(0..8):

%p q:= 2: count:= 0:

%p while count < 9 do

%p p:= q; q:= nextprime(q);

%p t:= p*q+p+q;

%p k:= padic:-ordp(t,5);

%p if V[k] = 0 and isprime(t/5^k) then

%p V[k]:= p; count:= count+1

%p fi

%p od:

%p convert(V,list);

%Y Cf. A000351, A002386, A126148.

%K nonn

%O 0,1

%A _Robert Israel_, Dec 25 2022

%E a(11) from _Michael S. Branicky_, Dec 26 2022

%E a(12)-a(15) from _David A. Corneth_, Dec 26 2022

%E a(16) from _Martin Ehrenstein_, Dec 27 2022

%E a(17)-a(21) from _Daniel Suteu_, Dec 28 2022