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A050414
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Numbers k such that 2^k - 3 is prime.
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40
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3, 4, 5, 6, 9, 10, 12, 14, 20, 22, 24, 29, 94, 116, 122, 150, 174, 213, 221, 233, 266, 336, 452, 545, 689, 694, 850, 1736, 2321, 3237, 3954, 5630, 6756, 8770, 10572, 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
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OFFSET
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1,1
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COMMENTS
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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
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
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LINKS
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MATHEMATICA
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Do[ If[ PrimeQ[ 2^n -3 ], Print[n]], { n, 1, 15000 }]
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PROG
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(PARI) for(n=2, 10^5, if(ispseudoprime(2^n-3), print1(n, ", "))) \\ Felix Fröhlich, Jun 23 2014
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(40)=20757, verified with 20 iterations of Miller-Rabin test, from Dmitry Kamenetsky, Jul 12 2008
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
a(59) and a(60) (corresponding to probable primes since they are PRP 3,5,7) discovered by Paul Bourdelais, Mar 26 2012
a(61) to a(63) correspond to probable primes discovered by Paul Bourdelais, Jun 18 2019
a(64) corresponds to a probable prime discovered by Paul Bourdelais, Jul 16 2019
a(65) corresponds to a probable prime discovered by Paul Bourdelais, Apr 20 2020
a(66) corresponds to a probable prime discovered by Paul Bourdelais, May 28 2020
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STATUS
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approved
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