1,2

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<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]

T. D. Noe, Table of n, a(n) for n=1..10000

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]

A014085 (number of primes between n^2 and (n+1)^2), A134034 (uses a larger K)

Sequence in context: A003649 A003650 A059233 this_sequence A101873 A079211 A081418

Adjacent sequences: A143895 A143896 A143897 this_sequence A143899 A143900 A143901

nice,nonn

T. D. Noe (noe(AT)sspectra.com), Sep 04 2008

Definition corrected. - T. D. Noe (noe(AT)sspectra.com), Sep 26 2008