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A339174
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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.
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1
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1, 1, 1, 2, 5, 9, 6, 79, 16, 219, 580, 387, 189, 7067, 1803, 6582, 31917, 18888, 20973, 132755, 11419, 50111
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OFFSET
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1,4
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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PROG
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(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
k, c, b = 1, 1, (a-1)*a
while True:
c += b
if isprime(c):
a = c
break
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CROSSREFS
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KEYWORD
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nonn,hard,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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