A230772 Number of primes in the half-open interval [n, 3*n/2).

0, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 3, 3, 3, 2, 3, 3, 3, 4, 5, 4, 4, 4, 4, 4, 4, 5, 5, 4, 4, 5, 6, 5, 5, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 6, 6, 7, 7, 7, 7, 7, 7, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 8, 8, 9, 9, 9, 9, 8, 8, 8, 8, 7, 8, 8, 8, 9, 9, 8, 8, 9, 10, 10, 10

Offset: 1

Author

Jean-Christophe Hervé, Oct 30 2013

Keywords

• nonn
• easy

Comments

Suggested by Bertrand's postulate (actually a theorem): for all x > 1, there is a prime number between x and 2x (see related references and links mentioned in A060715, A166968 and A143227).

For all n > 1, a(n)>=1 (that is, there is always a prime between n and 3*n/2); this can be seen using the stronger result proved by Jitsuro Nagura in 1952: for n >= 25, there is always a prime between n and (1 + 1/5)n.

Successive terms vary by no more than +/- one unit.

Examples

			a(29)=5 since five primes (29,31,37,41,43) are located between 29 and 43.5.
		

Links

• Jean-Christophe Hervé, Table of n, a(n) for n = 1..6667
• Jitsuro Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28, 1952, 177-181.

Crossrefs

Cf. A060715, A166968, A143227, A000720, A010051.

Programs

Mathematica

a[n_]:=PrimePi[Ceiling[1.5*n]-1]-PrimePi[n-1]; Table[a[n], {n, 2, 100}]

R

nvalues <- 1000
  A <- vector('numeric', nvalues)
  A[1] <- 0
  # primepi = table of the primepi sequence A000720
  for(i in 2:nvalues) A[i] <- primepi[ceiling(1.5*i)-1]-primepi[i-1]

Formula

a(n) = sum(A010051(n+k): 0<=k<3*n/2).

a(n) = A000720(ceiling(1.5*n)-1) - A000720(n-1).