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%I #73 Dec 02 2023 14:41:57
%S 3,4,5,6,9,10,12,14,20,22,24,29,94,116,122,150,174,213,221,233,266,
%T 336,452,545,689,694,850,1736,2321,3237,3954,5630,6756,8770,10572,
%U 14114,14400,16460,16680,20757,26350,30041,34452,36552,42689,44629,50474,66422,69337,116926,119324,123297,189110,241004,247165,284133,354946,394034,702194,750740,840797,1126380,1215889,1347744,1762004,2086750
%N Numbers k such that 2^k - 3 is prime.
%C With 65 known primes corresponding to k < 1762005, these primes appear to be more common than Mersenne primes. Of course at this time, the larger terms correspond only to probable primes. - _Paul Bourdelais_, Feb 04 2012
%C The numbers 2^k-3 and 2^k-1 are both primes for k = 3, 5, ? The lesser number 2^p-3 is prime for primes p = 3, 5, 29, 233, 42689, 69337, ... - _Thomas Ordowski_, Sep 18 2015
%C The terms a(43)-a(49) were found by Paul Underwood, a(50)-a(51) found by M. Frind and P. Underwood, a(52) found by Gary Barnes, a(53)-a(58) found by M. Frind and P. Underwood, and a(59)-a(66) found by Paul Bourdelais (see link Henri Lifchitz and Renaud Lifchitz). - _Elmo R. Oliveira_, Dec 02 2023
%H Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/ugradnumthy/squaresandinfmanyprimes.pdf">Square patterns and infinitude of primes</a>, University of Connecticut, 2019.
%H Henri Lifchitz and Renaud Lifchitz (Editors), <a href="http://www.primenumbers.net/prptop/searchform.php?form=2%5En-3">Search for 2^n-3</a>, PRP Top Records.
%e k = 22, 2^22 - 3 = 4194301 is prime.
%e k = 24, 2^24 - 3 = 16777213 is prime.
%t Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]
%o (PARI) for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ _Felix Fröhlich_, Jun 23 2014
%Y Cf. A045768, A050415, A057732 (numbers k such that 2^k + 3 is prime).
%Y Cf. Sequences of numbers k such that 2^k - d is prime: A000043 (d=1), A050414 (d=3), A059608 (d=5), A059609 (d=7), A059610 (d=9), A096817 (d=11), A096818 (d=13), A059612 (d=15), A059611 (d=17), A096819 (d=19), A096820 (d=21), A057220 (d=23), A356826 (d=29).
%K nonn
%O 1,1
%A _Jud McCranie_, Dec 22 1999
%E More terms from _Robert G. Wilson v_, Sep 15 2000
%E More terms from _Andrey V. Kulsha_, Feb 11 2001
%E a(40) verified with 20 iterations of Miller-Rabin test, from _Dmitry Kamenetsky_, Jul 12 2008
%E a(41) a new PRP term, from _Serge Batalov_, Oct 20 2008
%E Corrected and extended by including two smaller (apparently known) PRP and 16 larger terms from PRP Top Records of this form, all discovered by M. Frind & P. Underwood, _Gary Barnes_, Oct 20 2008
%E a(59)-a(60) discovered by _Paul Bourdelais_, Mar 26 2012
%E a(61)-a(63) discovered by _Paul Bourdelais_, Jun 18 2019
%E a(64) discovered by _Paul Bourdelais_, Jul 16 2019
%E a(65) discovered by _Paul Bourdelais_, Apr 20 2020
%E a(66) discovered by _Paul Bourdelais_, May 28 2020
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