A230022 - OEIS

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A230022

a(n) = |{the number of primes in the interval (k*n, (k+1)*n]: k = 0, 1, ..., n-1}|.

1, 1, 2, 2, 3, 3, 3, 4, 3, 4, 5, 5, 4, 5, 4, 5, 5, 6, 7, 6, 5, 6, 6, 6, 5, 5, 7, 6, 6, 7, 7, 6, 6, 7, 7, 8, 9, 8, 9, 9, 8, 8, 8, 9, 8, 9, 9, 8, 10, 10, 9, 10, 9, 10, 10, 10, 10, 11, 10, 10, 9, 10, 9, 11, 10, 11, 11, 11, 11, 11, 11, 11, 10, 12, 11, 10, 11, 12, 13, 11

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OFFSET

1,3

COMMENTS

Conjecture: (i) a(n) is at least sqrt(n-1) for each n > 0, and equality holds only when n is 2 or 26.

(ii) The sequence contains all positive integers.

We have verified part (i) of the conjecture for n up to 10000.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..4000

Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

EXAMPLE

a(1) = 1 since the interval (0,1*1] contains no prime, and the set {0} has cardinaly 1.