A117490 Number of primes between n and n^2 (with n and n^2 excluded).

0, 1, 2, 4, 6, 8, 11, 14, 18, 21, 25, 29, 33, 38, 42, 48, 54, 59, 64, 70, 77, 84, 90, 96, 105, 113, 120, 128, 136, 144, 151, 161, 170, 180, 189, 199, 207, 216, 228, 239, 250, 261, 269, 281, 292, 305, 314, 327, 342, 352, 363, 378, 393, 405, 418, 429, 441, 458, 470

Offset: 1

Author

Paolo P. Lava and Giorgio Balzarotti, Mar 22 2006

Keywords

• easy
• nonn

Comments

A famous Japanese mathematics book states that this sequence is nonzero (for n>1) if the Riemann Hypothesis is true, but this statement seems to be false.

If the n-th prime is denoted by p(n) then a(j) = number of nonzero values of floor (j^2/p(n)), over all n >= 1, (derived from A165974). - Christopher Hunt Gribble, Oct 03 2009

Examples

			For n = 5: between 5+1 = 6 and 5^2-1 = 24 there are the following six primes: 7, 11, 13, 17, 19, 23.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A010051, A014085, A038107, A060715, A073882, A117491, A165974.

Programs

Maple

P:=proc(n) local i,j,np; for i from 1 by 1 to n do np:=0; for j from i+1 by 1 to i^2-1 do if isprime(j) then np:=np+1; fi; od; print(np); od; end: P(100);

Mathematica

a[n_] := PrimePi[n^2 - 1] - PrimePi[n]; Array[a, 59] (* Robert G. Wilson v, Apr 06 2006 *)

Formula

a(n) = pi(n^2) - pi(n), cf. A000720.

a(n) = A038107(n) - A000720(n) = A073882(n) - A010051(n). - Reinhard Zumkeller, May 20 2010