A229989 Number of primes in the interval [floor(n/2), floor(3n/2)].

0, 2, 2, 3, 4, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 6, 7, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 13, 14, 14, 14, 15, 16, 16, 16, 17, 18, 18, 18

Offset: 1

Author

Clark Kimberling, Oct 09 2013

Keywords

• nonn
• easy

Comments

Conjectures:

(1) a(n+1) - a(n) = 1 for infinitely many n;

(2) a(n+1) - a(n) = -1 for infinitely many n;

(3) a(n+1) - a(n) = -1 if and only if n = 2*prime(m+1) - 1.

Examples

			a(5) = 4 counts the primes in the interval [2,7].
		

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A076274, A229990, A056172.

Programs

Maple

with(numtheory): A229989 := proc(n) return pi(floor((3/2)*n))-pi(floor(n/2)-1): end proc: seq(A229989(n), n=1..75); # Nathaniel Johnston, Oct 11 2013

Mathematica

z = 1000; c[n_] := PrimePi[Floor[3 n/2]] - PrimePi[Floor[n/2]-1];
t = Table[c[n], {n, 1, z}];            (* A229989 *)
Flatten[Position[Differences[t], -1]]  (* A076274? *)
Flatten[Position[Differences[t], 1]]   (* A229990 *)