A285786 Number of primes p with 2(n-1)^2 < p <= 2n^2.

1, 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: 1

Author

Ralf Steiner, Apr 26 2017

Keywords

• nonn

Comments

The author of the sequence conjectures that a(n) >= 1 for all n. This conjecture is similar to the famous conjecture made by Adrien-Marie Legendre that there is always a prime between n^2 and (n+1)^2, see A014085. - Antti Karttunen, May 01 2017

Examples

			For n = 1, the primes from 2*((1-1)^2) to 2*(1^2) (in semiopen range ]0, 2]) are: 2, thus a(1) = 1.
For n = 2, the primes from 2*((2-1)^2) to 2*(2^2) (in semiopen range ]2, 8]) are: 3, 5 and 7, thus a(2) = 3.
For n = 3, the primes from 2*((3-1)^2) to 2*(3^2) (in semiopen range ]8, 18]) are: 11, 13 and 17, thus a(3) = 3.
For n = 4, the primes from 2*((4-1)^2) to 2*(4^2) (in semiopen range ]18, 32]) are: 19, 23, 29 and 31, thus a(4) = 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001105, A000720 (number of primes), A014085 (between n^2 and (n+1)^2), A285738, A285388.

Programs

Maple

R:= [0, seq(numtheory:-pi(2*n^2),n=1..100)]:
R[2..-1] - R[1..-2]; # Robert Israel, May 01 2017

Mathematica

Table[Length[Select[FactorInteger[Numerator[Table[2^(1 - 2 n^2) n Binomial[2 n^2, n^2], {n, 1, k}]]][[k]][[All, 1]], # > 2 (k - 1)^2 &]], {k, 1, 60}]
Flatten[{1,2,Table[PrimePi[2 k^2] - PrimePi[2 (k - 1)^2], {k, 3, 60}]}]
(* Second program: *)
Array[PrimePi[2 #^2] - PrimePi[2 (# - 1)^2] &, 60] (* Michael De Vlieger, Apr 26 2017, at the suggestion of Robert G. Wilson v. *)

PARI

a(n) = (primepi(2*n^2)-primepi(2*(n-1)^2)) \\ David A. Corneth, Apr 27 2017, edited by Antti Karttunen, May 01 2017

PARI

a(n)=my(s); forprime(p=2*n^2 - 4*n + 3, 2*n^2, s++); s \\ Charles R Greathouse IV, May 10 2017

Python

from sympy import primepi
def a(n): return primepi(2*n**2) - primepi(2*(n - 1)**2) # Indranil Ghosh, May 01 2017

Formula

From Antti Karttunen, May 01 2017: (Start)

a(1) = 1, for n > 1, a(n) = A000720(A001105(n)) - A000720(A001105(n-1)).

For all n except n=2, a(n) <= n.

(End)

First differences of A278114: a(n) = A278114(n) - A278114(n-1) for n > 0, if we use A278114(0) = 0. A278114(n) = Sum_{k=1..n} a(n). - M. F. Hasler, May 02 2017

Extensions

Definition and value of a(2) changed by Antti Karttunen, May 01 2017