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A246119 a(n) is the least k such that k^(2^n)*(k^(2^n) - 1) + 1 is prime. 4
2, 2, 2, 5, 4, 2, 5, 196, 14, 129, 424, 484, 22, 5164, 7726, 13325, 96873, 192098, 712012, 123447 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
COMMENTS
Numbers of the form k^m*(k^m-1)+1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of A205506 with only m=2^n.
Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.
The numbers of this form are Generalized Unique primes (see Links section).
a(16)=96873 corresponds to a prime with 653552 decimal digits.
The search for a(17) which corresponds to a 1385044-decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.
a(18) <= 712012 corresponds to a prime with 3068389 decimal digits. - Serge Batalov, Jan 15 2018
a(19) <= 123447 corresponds to a prime with 5338805 decimal digits. - Serge Batalov, Jan 15 2018
a(20) <= 465859 corresponds to a prime with 11887192 decimal digits (not all lower candidates have been checked). This is the largest known non-Mersenne prime at the time of its discovery. - Serge Batalov, May 31 2023
LINKS
C. Caldwell, The Prime Pages, Phi_3(-465859^1048576)
FORMULA
a(n) = A085398(3*2^(n+1)). - Jinyuan Wang, Jan 01 2023
MATHEMATICA
Table[SelectFirst[Range@ 200, PrimeQ[#^(2^n) (#^(2^n) - 1) + 1] &], {n, 0, 9}] (* Michael De Vlieger, Jan 15 2018 *)
PROG
(PARI)
a(n)=k=1; while(!ispseudoprime(k^(2^n)*(k^(2^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++) \\ Derek Orr, Aug 14 2014
CROSSREFS
Sequence in context: A174577 A194684 A076737 * A210562 A208512 A208908
KEYWORD
nonn,more,hard
AUTHOR
Serge Batalov, Aug 14 2014
EXTENSIONS
a(16) from Serge Batalov, Dec 30 2014
a(17) from Serge Batalov, Feb 10 2015
a(18-19) from Serge Batalov, May 31 2023
STATUS
approved

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Last modified June 17 15:57 EDT 2024. Contains 373463 sequences. (Running on oeis4.)