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%I #47 Jul 08 2023 21:06:50
%S 11,21,127,149,469,2019,21689,25679
%N Numbers k such that 17^k - 16 is prime.
%C Related to hyperperfect numbers of a certain form.
%C From Daniel Minoli (daniel.minoli(AT)ses.com), Aug 27 2009: (Start)
%C Minoli defined the sequences and concepts that follow in the 1980 IEEE paper below:
%C - For t=2 to infinity, the sequence m(n,t) = n exp(t) - (n-1) is called a Mersenne Sequence Rooted on n
%C - If n is prime, this sequence is called a Legitimate Mersenne Sequence
%C - Any j belonging to the sequence m(n,t) is called a Generalized Mersenne Number (n-GMN)
%C - If j belonging to the sequence m(n,t) is prime, it is then called a n-Generalized Mersenne Prime (n-GMP).
%C Note: m(n,t) = n*m(n,t-1) + n exp(2) - 2*n+1.
%C These numbers play a role in the context of hyperperfect numbers.
%C (End)
%C The next terms are > 4000. - _Vincenzo Librandi_, Sep 27 2012
%C a(7)=21689 and a(8)=25679 correspond to probable primes, found with Dario Alpern's factorization tool using the elliptic curve method; no more terms < 35000. - _Andrej Jakobcic_, Feb 17 2019
%D Daniel Minoli, Voice over MPLS, McGraw-Hill, New York, NY, 2002, ISBN 0-07-140615-8 (pp. 114-134).
%H Dario Alejandro Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method</a>
%H J. S. McCranie, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/mccranie.html">A study of hyperperfect numbers</a>, J. Int. Seqs. Vol. 3 (2000) #P00.1.3.
%H Daniel Minoli and Robert Bear, <a href="https://www.pme-math.org/journal/issues/PMEJ.Vol.6.No.3.pdf">Hyperperfect Numbers</a>, Pi Mu Epsilon Journal, Fall 1975, pp. 153-157.
%H Daniel Minoli, W. Nakamine, <a href="https://dx.doi.org/10.1109/ICASSP.1980.1170906">Mersenne Numbers Rooted On 3 For Number Theoretic Transforms</a>, 1980 IEEE International Conf. on Acoust., Speech and Signal Processing.
%t Select[Range[3000], PrimeQ[17^# - 16] &] (* _Vincenzo Librandi_, Sep 27 2012 *)
%o (PARI) isok(n) = isprime(17^n-16); \\ _Michel Marcus_, Mar 11 2016
%K nonn,more
%O 1,1
%A _Jud McCranie_
%E a(6) from _Vincenzo Librandi_, Sep 27 2012
%E a(7) and a(8) from _Andrej Jakobcic_, Feb 17 2019
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