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%I #31 Jul 25 2023 10:58:40
%S 1,3,5,15,5,59,159,189,569,105,1557,2549,2439,13797,25353,5627,24317,
%T 231425,164073
%N Ultra-useful primes: smallest k such that 2^(2^n) - k is prime.
%C 2^(2^16) - 5627 was found by Joel Levy in Feb 2004. - _Donovan Johnson_, Sep 13 2008
%C Corresponding numbers to entries a(1) to a(12) are proven primes, higher terms are probable primes. - _Matthias Baur_, Mar 17 2020
%H Rosetta Code, <a href="https://rosettacode.org/wiki/Ultra_useful_primes">Ultra-useful primes</a>.
%F a(n) = A013603(2^n). - _Jinyuan Wang_, Jun 06 2020
%e For n = 3, we see that 2^(2^3) = 2^8 = 256, which is clearly not prime.
%e 256 - 1 = 255 = 3 * 5 * 17, so a(3) is not 1.
%e 256 - 2 = 254 = 2 * 127, so a(3) is not 2 either.
%e 256 - 3 = 253 = 11 * 23, so a(3) is not 3 either.
%e 256 - 5 = 251, which is prime, so a(3) = 5.
%t ultraUseful[n_] := Module[{x = 2^(2^n)}, x - NextPrime[x, -1]]; Array[ultraUseful, 17] (* _Harvey P. Dale_, Jun 04 2011 *)
%Y Cf. A013603, A058221.
%K nonn,hard,nice,more
%O 1,2
%A _Warren D. Smith_, Nov 30 2000
%E a(16)-a(17) from _Donovan Johnson_, Sep 13 2008
%E a(18) from _Matthias Baur_, Mar 17 2020
%E a(19) from _Matthias Baur_, Apr 04 2020
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