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%I #7 Feb 23 2020 21:19:54
%S 3,7,31,103,127,337,1663,5407,8191,131071,346111,524287,2970343,
%T 3655807,22151167,109051903,617831551,1631461441,2007952543,
%U 2147483647,32127240703,194664464383,275716983697,958348132351,1357375919743,1670616516607,49834102882303,57349132609183
%N Primes of the form 2^r * 13^s - 1.
%C s = 0 is "trivial" case of Mersenne primes: 3, 7, 31, 127, 8191, 131071, 524287, 2147483647, ...
%C Mersenne prime exponents r: 2, 3, 5, 7, 13, 17, 19, 31, ...
%C Necessarily r odd as for r = 2*k and p a prime of form 6*n+1: 2^(2*k) * p^j - 1 a multiple of 3.
%C Proof by induction with 2^2 * p^1 - 1 = 4*(6*n+1) - 1 = 3*(8*n+1), 2^2(k+1) * p^j - 1 = 4* (2^k * p^j - 1) + 3.
%C No prime in case i = j = k (k>1) as a^k - 1 has divisor a - 1.
%D Peter Bundschuh: Einfuehrung in die Zahlentheorie, Springer-Verlag GmbH Berlin, 2002)
%D Leonard E. Dickson: History of the Theory of numbers, vol. I, Dover Publications, 2005
%D Paulo Ribenboim, Wilfrid Keller, Joerg Richstein: Die Welt der Primzahlen, Springer-Verlag GmbH Berlin, 2006
%H Jinyuan Wang, <a href="/A173062/b173062.txt">Table of n, a(n) for n = 1..950</a>
%e 2^2*13^0 - 1 = 3 = prime(2) => a(1).
%e 2^3*13^1 - 1 = 103 = prime(27) => a(4).
%e 2^7*13^9 - 1 = 1357375919743 = prime(50467169414) => a(25).
%e list of (r,s) pairs: (2,0), (3,0), (5,0), (3,1), (7,0), (1,2), (7,1), (5,2), (13,0), (17,0), (11,2), (19,0), (3,5), (7,4), (17,2), (23,1), (7,6), (1,8), (5,7), (31,0), (9,7), (19,5), (1,10), (25,4), (7,9), (11,8), (27,5), (5,11), (25,6), (19,8), (13,10), (3,13), (29,7), (5,14), (39,5), (15,13), (5,16), ...
%o (PARI) lista(nn) = {my(q=1/2, p, w=List([])); for(r=0, logint(nn, 2), q=2*q; p=q/13; for(s=0, logint(nn\q, 13), p=13*p; if(ispseudoprime(p-1), listput(w, p-1)))); Set(w); } \\ _Jinyuan Wang_, Feb 23 2020
%Y Cf. A000668, A005105, A077313, A077314, A077315.
%K nonn
%O 1,1
%A Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Feb 09 2010
%E a(26)-a(28) from _Jinyuan Wang_, Feb 23 2020
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