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.

1, 1, 2, 4, 8, 16, 5, 360, 142, 104, 34, 1904, 3127, 253, 1219, 8755, 16222, 7672, 22515

Offset: 1

Author

Rashid Naimi, Apr 10 2019

Keywords

• nonn
• hard
• more

Comments

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.

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.

Examples

			For p = 3, the smallest k for which f(k) = k*(p-1)*p-1 is prime is 1:
f(1) = k*(p-1)*p-1 = 1*(3-1)*3-1 = 5.
This sets p = 5 for the next iteration for which the smallest k for which f(k) is prime is 1:
f(1) = k*(p-1)*p-1 = 1*(5-1)*5-1 = 19.
This sets p = 19 for the next iteration for which the smallest k for which f(k) is prime is 2:
f(2) = k*(p-1)*p-1 = 2*(19-1)*19-1 = 683.
This sets p = 683 for the next iteration for which the smallest k for which f(k) is prime is 4:
f(4) = k*(p-1)*p-1 = 4*(683-1)*683-1 = 1863223.
This sets p = 1863223 for the next iteration for which the smallest k for which f(k) is prime is 8:
f(8) = k*(p-1)*p-1 = 8*(1863223-1)*1863223-1 = P14.
		

Crossrefs

Cf. A000058.

Programs

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))

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")

Formula

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.

Extensions

Definition clarified by Charlie Neder, Jun 03 2019

a(17) from Rashid Naimi, Aug 23 2019

a(18) from Rashid Naimi, Oct 22 2019

a(19) from Rashid Naimi, Aug 01 2020