

A084712


Smallest prime of the form (2n)^k +1, or 0 if no such number exists.


8



3, 5, 7, 0, 11, 13, 197, 17, 19, 401, 23, 577, 677, 29, 31, 0, 1336337, 37, 0, 41, 43, 197352587024076973231046657, 47, 5308417, 0, 53, 2917, 3137, 59, 61, 0, 0, 67, 0, 71, 73, 5477, 1238846438084943599707227160577, 79, 40960001, 83, 7057, 0, 89
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OFFSET

1,1


COMMENTS

It has not been proved that a(19), a(25), a(31), a(34), a(43) and a(46) are 0; if these values do exist, they have > 4000 digits. The other zeros are definite.  David Wasserman, Jan 03 2005
a((p1)/2) = p for primes p>2, or a(n) = 2n+1 for n = (p1)/2. All other positive a(n) belong to A002496 = primes of form n^2 + 1. Corresponding positive exponents k are powers of 2. They are listed in A079706.  Alexander Adamchuk, Sep 17 2006
Because k must be a power of 2, numbers of the form (2n)^k+1 are called generalized Fermat numbers with base 2n. These numbers, like the regular Fermat numbers, are seldom prime. I checked n=19, 25, 31, 34, 43, 46 with k up to 2^16 without finding any primes.  T. D. Noe, May 13 2008


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..500


EXAMPLE

a(7) = 197 = 14^2 +1 as 14 +1 =15 is not a prime.


MATHEMATICA

Table[k=1; While[p=1+(2n)^k; k<1024 && !PrimeQ[p], k=2k]; If[k==1024, 0, p], {n, 44}] (* T. D. Noe, May 13 2008 *)


CROSSREFS

Cf. A084713, A005097.
Sequence in context: A084713 A162538 A324990 * A031100 A031057 A133069
Adjacent sequences: A084709 A084710 A084711 * A084713 A084714 A084715


KEYWORD

nonn


AUTHOR

Amarnath Murthy, Jun 10 2003


EXTENSIONS

More terms from David Wasserman, Jan 03 2005


STATUS

approved



