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

1,4

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.

Links

Table of n, a(n) for n=1..22.

The 5000 Largest Known Primes, P587124 and P1174253.

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.

Example

[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.

Prog

(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

Crossrefs

Cf. A359340, A306601, A000058.

Sequence in context: A175640 A204913 A198455 * A018878 A021389 A198429

Adjacent sequences: A339171 A339172 A339173 * A339175 A339176 A339177

Keyword

nonn,hard,more

Author

Rashid Naimi, Nov 25 2020

Extensions

a(22) from Rashid Naimi, Jan 13 2023

Status

approved