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Search: id:A143935
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| A143935 |
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Number of primes between n^K and (n+1)^K, inclusive, where K=1.74717117169304146332. |
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+0
7
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| 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
(list; graph; listen)
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OFFSET
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1,1
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COMMENT
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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. [From Enoch Haga (Enokh(AT)comcast.net), Sep 24 2008]
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..10000
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MATHEMATICA
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k= 1.74717117169304146332; Table[Length[Select[Range[Ceiling[n^k], Floor[(n+1)^k]], PrimeQ]], {n, 150}]
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CROSSREFS
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A014085 (number of primes between n^2 and (n+1)^2)
Sequence in context: A085035 A083023 A084359 this_sequence A008616 A097471 A025868
Adjacent sequences: A143932 A143933 A143934 this_sequence A143936 A143937 A143938
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Sep 05 2008
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EXTENSIONS
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Corrected a(15) from 1 to 0 Enoch Haga (Enokh(AT)comcast.net), 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 (noe(AT)sspectra.com), Sep 25 2008
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