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 A143898 Number of primes between n^K and (n+1)^K, where K = log(1151)/log(95). 8
 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 1, 2, 2, 3, 2, 3, 3, 1, 4, 2, 3, 2, 1, 3, 2, 3, 2, 2, 2, 4, 1, 4, 2, 2, 2, 2, 3, 2, 3, 2, 4, 3, 2, 3, 3, 3, 3, 1, 3, 3, 2, 3, 3, 2, 3, 5, 3, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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 to (n+1)^K for n>0. This value of K was found using exact interval arithmetic. For each n <= 110 and for each prime p in the range n to n^1.7, 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 intervals. The least value in those intervals is K, which is log(1151)/log(95). We computed 10^5 terms of this sequence to give us confidence that a(n)>0 for all n. More details about the algorithm: The n^1.7 limit was chosen because we were fairly certain that K would be less than 1.7. Let k(n) be the union of the intervals k(n,p) for p 0 with no prime p satisfying n^e <= p < (n+1)^e. - Peter Munn, Mar 02 2017 The author's description of the calculation of K implies that K is not an isolated qualifying value; equivalently that K is also the least real value for which there is a positive epsilon such that for all exponent e, K <= e <= K+epsilon and integer n > 0 there is a prime p satisfying n^e <= p <= (n+1)^e. This is a necessary precondition for my Mar 02 2017 deduction from the author's conjecture. - Peter Munn, Aug 21 2019 LINKS T. D. Noe, Table of n, a(n) for n = 1..10000 MATHEMATICA k= 1.547777108714197624815033; Table[Length[Select[Range[Ceiling[n^k], Floor[(n+1)^k]], PrimeQ]], {n, 150}] (* T. D. Noe, Sep 08 2008 *) CROSSREFS A014085 (number of primes between n^2 and (n+1)^2), both A134034 and A143935 use a larger K. Sequence in context: A003650 A059233 A327924 * A332636 A238747 A101873 Adjacent sequences:  A143895 A143896 A143897 * A143899 A143900 A143901 KEYWORD nice,nonn AUTHOR T. D. Noe, Sep 04 2008, Sep 26 2009, Oct 21 2009 EXTENSIONS Removed some comments which changed the definition of this sequence. - N. J. A. Sloane, Oct 21 2009 STATUS approved

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Last modified April 21 15:39 EDT 2021. Contains 343154 sequences. (Running on oeis4.)