A285269 Number of (odd) primes between 2*n^2 and 2*(n+1)^2.

0, 3, 3, 4, 4, 5, 5, 6, 6, 9, 7, 8, 7, 9, 10, 10, 9, 12, 10, 11, 13, 11, 14, 13, 14, 13, 14, 16, 16, 15, 15, 16, 17, 18, 19, 14, 22, 19, 18, 16, 22, 18, 24, 20, 22, 22, 20, 23, 24, 22, 23, 21, 25, 27, 24, 27, 26, 25, 27, 25, 23, 33, 28, 25, 29, 28, 31, 30, 33, 29

Offset: 0

Author

M. F. Hasler, May 02 2017

Keywords

• nonn

Comments

Essentially the same as A285786, except for the offset and initial values.

Examples

			a(0) = 0 because between 2*0^2 = 0 and 2*1^2 = 2, there are no (odd) primes.
a(1) = 3 because between 2*1^2 = 2 and 2*2^2 = 8, there are the 3 (odd) primes 3, 5 and 7.
a(2) = 3 because between 2*2^2 = 8 and 2*3^2 = 18, there are the 3 primes 11, 13 and 17.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000

Programs

Mathematica

Join[{0}, Differences[PrimePi[2*Range[100]^2]]] (* Paolo Xausa, Nov 22 2024 *)

PARI

a(n)=primepi(2*(n+1)^2-1)-primepi(2*n^2)

Formula

a(n) = A285786(n+1), for all n >= 2.

First differences of A278114, a(n) = A278114(n+1) - A278114(n), for n > 0.