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

Offset: 1

Author

Zhi-Wei Sun, Feb 23 2014

Keywords

• nonn

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.

Examples

			a(1) = 1 since the interval (0,1*1] contains no prime, and the set {0} has cardinaly 1.
a(3) = 2 since the intervals (0, 1*3], (1*3, 2*3], (2*3, 3*3] contain exactly 2, 1, 1 primes respectively, and the set {2, 1, 1} has cardinality 2.
		

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

Crossrefs

Cf. A000040, A000720, A238277, A238278, A238281.

Programs

Mathematica

d[k_,n_]:=PrimePi[(k+1)*n]-PrimePi[k*n]
a[n_]:=Length[Union[Table[d[k,n],{k,0,n-1}]]]
Table[a[n],{n,1,80}]