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

%I A143935 #30 Sep 30 2021 22:43:17
%S A143935 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,
%T A143935 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,
%U A143935 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
%N A143935 Number of primes between n^K and (n+1)^K, inclusive, where K=log(127)/log(16).
%C A143935 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.
%C A143935 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
%C A143935 It suffices to check members of floor(A002386^(1/k)). - _Charles R Greathouse IV_, Feb 03 2011
%C A143935 The constant log(127)/log(16) is A194361. - _John W. Nicholson_, Dec 13 2013
%H A143935 T. D. Noe, <a href="/A143935/b143935.txt">Table of n, a(n) for n = 1..10000</a>
%H A143935 Carlos Rivera, <a href="http://www.primepuzzles.net/conjectures/conj_060.htm">Conjecture 60: Generalization of Legendre's Conjecture</a>
%t A143935 k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k],Floor[(n+1)^k]], PrimeQ]], {n,150}]
%t A143935 With[{k=Log[16,127]},Table[Count[Range[Ceiling[n^k],Floor[(n+1)^k]],_?PrimeQ],{n,110}]] (* _Harvey P. Dale_, Apr 03 2019 *)
%Y A143935 Cf. A014085 (number of primes between n^2 and (n+1)^2).
%Y A143935 Cf. A144256, A194361.
%K A143935 nonn
%O A143935 1,1
%A A143935 _T. D. Noe_, Sep 05 2008
%E A143935 Corrected a(15) from 1 to 0 _Enoch Haga_, Sep 24 2008
%E A143935 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