A227066 Number of primes between n and 2n inclusive equals the number of primes between 2n and 3n inclusive.

5, 6, 15, 21, 25, 26, 28, 29, 30, 33, 35, 38, 39, 44, 47, 50, 62, 63, 65, 67, 74, 78, 80, 86, 94, 105, 108, 109, 112, 114, 153, 155, 164, 165, 170, 171, 172, 173, 174, 207, 208, 209, 215, 276, 279, 282, 283, 285, 287, 293, 294, 297, 298, 299, 504, 704, 712, 714, 1308, 1316, 1322, 1328

Offset

1,1

Comments

There are almost certainly no more terms. Can anyone prove this conclusively?

The analogous sequence where the number of primes between 0 and n equals the number of primes from n to 2n is finite, consisting of only 2, 4, and 10. See Ehrhart. - Charles R Greathouse IV, Jul 02 2013

Links

Table of n, a(n) for n=1..62.

Eugene Ehrhart, On prime numbers, Fibonacci Quarterly 26:3 (1988), pp. 271-274.

Formula

Numbers n such that pi(3n) + pi(n-1) = 2pi(2n).

Example

4 is not in the sequence since the interval [n,2n] = [4,8] contains two primes (5 and 7), while the interval [2n,3n] = [8,12] contains only one prime (11).
6 is in the sequence since the intervals [6,12] contains two primes (7 and 11), and the interval [12,18] also contains two primes (13 and 17).

Mathematica

Do[If[PrimePi[3n]+PrimePi[n-1]==2*PrimePi[2n], Print[n]], {n, 4, 10^6}]

Prog

(PARI) is(n)=my(t=primepi(3*n)+primepi(n-1)); t%2==0 && t==2*primepi(2*n) \\ Charles R Greathouse IV, Jul 02 2013

Crossrefs

Sequence in context: A121847 A030533 A329634 * A115908 A247962 A241307

Adjacent sequences: A227063 A227064 A227065 * A227067 A227068 A227069

Keyword

nonn

Author

Zak Seidov, Jun 30 2013

Status

approved