OFFSET
1,1
COMMENTS
This value of K is conjectured to be the least possible such that there is at least one prime in the range n^k and (n+1)^k for all n>0 and k>=K. This value of K was found using exact interval arithmetic. For each n <= 300 and for each prime p in the range n to n^2, we computed an interval k(n,p) such that p is between n^k(n,p) and (n+1)^k(n,p). The intersection of all these intervals produces a list of 29 intervals. The last interval appears to be semi-infinite beginning with K, which is log(127)/log(16). See A143898 for the smallest number in the first interval.
My UBASIC program indicates no prime between 113.457 ... and 126.999 .... Next prime > 113 is 127. I would like someone to check this. - Enoch Haga, Sep 24 2008
It suffices to check members of floor(A002386^(1/k)). - Charles R Greathouse IV, Feb 03 2011
The constant log(127)/log(16) is A194361. - John W. Nicholson, Dec 13 2013
LINKS
T. D. Noe, Table of n, a(n) for n = 1..10000
Carlos Rivera, Conjecture 60: Generalization of Legendre's Conjecture
MATHEMATICA
k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k], Floor[(n+1)^k]], PrimeQ]], {n, 150}]
With[{k=Log[16, 127]}, Table[Count[Range[Ceiling[n^k], Floor[(n+1)^k]], _?PrimeQ], {n, 110}]] (* Harvey P. Dale, Apr 03 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Sep 05 2008
EXTENSIONS
Corrected a(15) from 1 to 0 Enoch Haga, Sep 24 2008
My intention was to include the endpoints of the range. Using k=log(127)/log(16), the endpoint for n=15 is exactly 127, which is prime. - T. D. Noe, Sep 25 2008
STATUS
approved