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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, 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
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
KEYWORD
nonn,hard,more
AUTHOR
Rashid Naimi, Nov 25 2020
EXTENSIONS
a(22) from Rashid Naimi, Jan 13 2023
STATUS
approved