A339174 Let b(1) = 2 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, 1, 2, 5, 9, 6, 79, 16, 219, 580, 387, 189, 7067, 1803, 6582, 31917, 18888, 20973, 132755, 11419, 50111

Offset: 1

Author

Rashid Naimi, Nov 25 2020

Keywords

• nonn
• hard
• more

Comments

The corresponding primes in order are 3, 7, 43, 3613, 65250781, P17, P34, P70, P141, P284, P571, P1144, P2290, P4584, P9170, P18344, P36692, P73387, P146778, P293560, P587124, P1174253.

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

			[Corrected by _Peter Munn_, Nov 05 2022]
For p = 2, the smallest k for which f(k) = k*(p-1)*p+1 is prime is 1 because we have: f(1) = k*(p-1)*p+1 = 1*(2-1)*2+1 = 3.
This sets p = 3 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*(3-1)*3+1 = 7.
		

Links

• The 5000 Largest Known Primes, P587124 and P1174253.

Crossrefs

Cf. A359340, A306601, A000058.

Programs

PARI

my(p=2, k=1); while(1, my(runningP=k*(p-1)*p+1); if(ispseudoprime(runningP), print1(k, ", "); k=1; p=runningP; , k=k+1))

PARI

my(k=[1, 1, 1, 2, 5, 9, 6, 79, 16, 219, 580, 387, 189, 7067, 1803, 6582, 31917, 18888, 20973, 132755, 11419, 50111], p=2); for(i=1, #k, p=k[i]*(p-1)*p+1); print("\n", p, "\n"); \\ to produce the P587124 prime

Python

from sympy import isprime
A339174_list, a = [2], 2
while len(A339174_list) < 10:
    k, c, b = 1, 1, (a-1)*a
    while True:
        c += b
        if isprime(c):
            A339174_list.append(k)
            a = c
            break
        k += 1 # Chai Wah Wu, Dec 04 2020

Formula

Nested f(k) = k*(p-1)*p+1 for p=2. 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

a(22) from Rashid Naimi, Jan 13 2023