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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 (list; graph; refs; listen; history; text; internal format)
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((p-1)/2) = p for primes p > 2, or a(n) = 2n+1 for n = (p-1)/2. All other positive a(n) belong to A002496 = primes of form m^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
Comments from N. J. A. Sloane, Jan 27 2024: (Start)
As pointed out by Max Alekseyev, the previous version violated the OEIS rules, since a(19) has not been confirmed. I therefore removed the terms starting at a(19).
The previous DATA line read:
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
The old b-file has been changed to an a-file.
(End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 1..500 (currently known terms, may contain 0s in place of unknown terms)
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
Sequence in context: A084713 A162538 A324990 * A371144 A031100 A031057
KEYWORD
nonn,more
AUTHOR
Amarnath Murthy, Jun 10 2003
EXTENSIONS
More terms from David Wasserman, Jan 03 2005
Edited by N. J. A. Sloane, Jan 27 2024 at the suggestion of Max Alekseyev
STATUS
approved

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Last modified June 24 06:40 EDT 2024. Contains 373663 sequences. (Running on oeis4.)