A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).

2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 5, 1, 4, 2, 3, 3, 4, 1, 5, 3, 3, 4, 3, 3, 3, 4, 4, 3, 5, 4, 3, 5, 2, 4, 5, 4, 5, 5, 3, 5, 5, 2, 6, 5, 4, 4, 4, 5, 5, 7, 5, 5, 3, 5, 6, 3, 8, 3, 4, 5, 6, 7, 5, 6, 8, 5, 4, 6, 6, 3, 7, 5, 4, 8, 5, 8, 6, 3, 7, 7, 6, 8, 7, 4, 5, 6, 5, 9, 9, 7, 6, 6, 6, 6, 7, 6, 4, 8, 5, 8, 8, 4

Offset: 1

Author

T. D. Noe, Sep 05 2008

Keywords

• nonn

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

Crossrefs

Cf. A014085 (number of primes between n^2 and (n+1)^2).

Cf. A144256, A194361.

Programs

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 *)

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