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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, 1, 2, 4, 8, 16, 5, 360, 142, 104, 34, 1904, 3127, 253, 1219, 8755, 16222, 7672, 22515
OFFSET
1,3
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.
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.
EXAMPLE
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.
PROG
(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))
The largest prime (P242682) can be generated by using the code:
(PARI) 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")
CROSSREFS
Cf. A000058.
Sequence in context: A178170 A127824 A088975 * A237851 A167425 A212638
KEYWORD
nonn,hard,more
AUTHOR
Rashid Naimi, Apr 10 2019
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
STATUS
approved