

A058220


Ultrauseful primes: smallest k such that 2^(2^n)  k is prime.


3



1, 3, 5, 15, 5, 59, 159, 189, 569, 105, 1557, 2549, 2439, 13797, 25353, 5627, 24317, 231425, 164073
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OFFSET

1,2


COMMENTS

2^(2^16)  5627 was found by Joel Levy in Feb 2004.  Donovan Johnson, Sep 13 2008
Corresponding numbers to entries a(1) to a(12) are proven primes, higher terms are probable primes.  Matthias Baur, Mar 17 2020


LINKS

Table of n, a(n) for n=1..19.


FORMULA

a(n) = A013603(2^n).  Jinyuan Wang, Jun 06 2020


EXAMPLE

For n = 3, we see that 2^(2^3) = 2^8 = 256, which is clearly not prime.
256  1 = 255 = 3 * 5 * 17, so a(3) is not 1.
256  2 = 254 = 2 * 127, so a(3) is not 2 either.
256  3 = 253 = 11 * 23, so a(3) is not 3 either.
256  5 = 251, which is prime, so a(3) = 5.


MATHEMATICA

ultraUseful[n_] := Module[{x = 2^(2^n)}, x  NextPrime[x, 1]]; Array[ultraUseful, 17] (* Harvey P. Dale, Jun 04 2011 *)


CROSSREFS

Cf. A013603, A058221.
Sequence in context: A270688 A270453 A270464 * A100181 A180620 A336882
Adjacent sequences: A058217 A058218 A058219 * A058221 A058222 A058223


KEYWORD

nonn,hard,nice,more


AUTHOR

Warren D. Smith, Nov 30 2000


EXTENSIONS

a(16)a(17) from Donovan Johnson, Sep 13 2008
a(18) from Matthias Baur, Mar 17 2020
a(19) from Matthias Baur, Apr 04 2020


STATUS

approved



