%I
%S 1,1,2,4,8,16,5,360,142,104,34,1904,3127,253,1219,8755,16222,7672,
%T 22515
%N Let b(1) = 3 and let b(n+1) be the least prime expressible as k*(b(n)1)*b(n))1; this sequence gives the values of k in order.
%C The corresponding primes in order are 5, 19, 683, 1863223, P14, P29, P57, P117, P235, P472, P945, P1893, P3789, P7581, P15164, P30332, P60668, P121339, P242682.
%C After each iteration the number of decimal digits is roughly twice that of the previous iteration. These primes can generally be easily certified using the N+1 method since all the prime factors for N+1 are known.
%F Nested f(k) = k*(p1)*p1 for p=3. After each iteration the last obtained f(k) is substituted for p. The primes can be certified using OpenPFGW by adding each previous iteration to the helper file.
%e For p = 3, the smallest k for which f(k) = k*(p1)*p1 is prime is 1:
%e f(1) = k*(p1)*p1 = 1*(31)*31 = 5.
%e This sets p = 5 for the next iteration for which the smallest k for which f(k) is prime is 1:
%e f(1) = k*(p1)*p1 = 1*(51)*51 = 19.
%e This sets p = 19 for the next iteration for which the smallest k for which f(k) is prime is 2:
%e f(2) = k*(p1)*p1 = 2*(191)*191 = 683.
%e This sets p = 683 for the next iteration for which the smallest k for which f(k) is prime is 4:
%e f(4) = k*(p1)*p1 = 4*(6831)*6831 = 1863223.
%e This sets p = 1863223 for the next iteration for which the smallest k for which f(k) is prime is 8:
%e f(8) = k*(p1)*p1 = 8*(18632231)*18632231 = P14.
%o (PARI) p=3; k=1; while(1, runningP=k*(p1)*p1; if(ispseudoprime(runningP), print1(k,", "); k=1; p=runningP;, k=k+1))
%o The largest prime (P242682) can be generated by using the code:
%o (PARI) k=[1, 1, 2, 4, 8, 16, 5, 360, 142, 104, 34, 1904, 3127, 253, 1219, 8755, 16222, 7672, 22515]; p=3; for(i=1, #k, p=k[i]*(p1)*p1); print("\n", p, "\n")
%Y Cf. A307334, A307498, A307499, A000058.
%K nonn,hard,more
%O 1,3
%A _Rashid Naimi_, Apr 10 2019
%E Definition clarified by _Charlie Neder_, Jun 03 2019
%E a(17) from _Rashid Naimi_, Aug 23 2019
%E a(18) from _Rashid Naimi_, Oct 22 2019
%E a(19) from _Rashid Naimi_, Aug 01 2020
