Search: id:A143898
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%I A143898
%S A143898 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,
%T A143898 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,
%U A143898 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
%N A143898 Number of primes between n^K and (n+1)^K, where K=1.547777108714197624815033.
%C A143898 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.
%C A143898 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<n^1.7. Then k(n) is the set 
               of exponents e such that the range n^e to (n+1)^e always contains 
               a prime. Let k be the intersection of all the k(n) intervals for 
               n up to N. Then k is the set of exponents e such that there is always 
               a prime in the range n^e to (n+1)^e for n<=N. The number K is the 
               least number in the set k. It appears that as N becomes larger, the 
               set k converges. See A143935. [From T. D. Noe (noe(AT)sspectra.com), 
               Sep 08 2008]
%H A143898 T. D. Noe, <a href="b143898.txt">Table of n, a(n) for n=1..10000</a>
%t A143898 k= 1.547777108714197624815033; Table[Length[Select[Range[Ceiling[n^k],
               Floor[(n+1)^k]], PrimeQ]], {n,150}] [From T. D. Noe (noe(AT)sspectra.com), 
               Sep 08 2008]
%Y A143898 A014085 (number of primes between n^2 and (n+1)^2), A134034 (uses a larger 
               K)
%Y A143898 Sequence in context: A003649 A003650 A059233 this_sequence A101873 A146289 
               A079211
%Y A143898 Adjacent sequences: A143895 A143896 A143897 this_sequence A143899 A143900 
               A143901
%K A143898 nice,nonn
%O A143898 1,2
%A A143898 T. D. Noe (noe(AT)sspectra.com), Sep 04 2008, Sep 26 2009, Oct 21 2009
%E A143898 Removed some comments which changed the definition of this sequence. 
               - N. J. A. Slaone, Oct 21 2009

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