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A306601 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.

%I A306601 #95 Aug 17 2025 12:42:46
%S A306601 1,1,2,4,8,16,5,360,142,104,34,1904,3127,253,1219,8755,16222,7672,
%T A306601 22515
%N A306601 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 A306601 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 A306601 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 A306601 Nested f(k) = k*(p-1)*p-1 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 A306601 For p = 3, the smallest k for which f(k) = k*(p-1)*p-1 is prime is 1:
%e A306601 f(1) = k*(p-1)*p-1 = 1*(3-1)*3-1 = 5.
%e A306601 This sets p = 5 for the next iteration for which the smallest k for which f(k) is prime is 1:
%e A306601 f(1) = k*(p-1)*p-1 = 1*(5-1)*5-1 = 19.
%e A306601 This sets p = 19 for the next iteration for which the smallest k for which f(k) is prime is 2:
%e A306601 f(2) = k*(p-1)*p-1 = 2*(19-1)*19-1 = 683.
%e A306601 This sets p = 683 for the next iteration for which the smallest k for which f(k) is prime is 4:
%e A306601 f(4) = k*(p-1)*p-1 = 4*(683-1)*683-1 = 1863223.
%e A306601 This sets p = 1863223 for the next iteration for which the smallest k for which f(k) is prime is 8:
%e A306601 f(8) = k*(p-1)*p-1 = 8*(1863223-1)*1863223-1 = P14.
%o A306601 (PARI) p=3; k=1; while(1, runningP=k*(p-1)*p-1; if(ispseudoprime(runningP), print1(k,", "); k=1; p=runningP;, k=k+1))
%o A306601 (PARI) /* The largest prime (P242682) can be generated by using the code: */ 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]*(p-1)*p-1); print("\n", p, "\n")
%Y A306601 Cf. A000058.
%K A306601 nonn,hard,more
%O A306601 1,3
%A A306601 _Rashid Naimi_, Apr 10 2019
%E A306601 Definition clarified by _Charlie Neder_, Jun 03 2019
%E A306601 a(17) from _Rashid Naimi_, Aug 23 2019
%E A306601 a(18) from _Rashid Naimi_, Oct 22 2019
%E A306601 a(19) from _Rashid Naimi_, Aug 01 2020