A290564 Number of primes between n^2 and 2*n^2.

1, 2, 3, 5, 6, 9, 10, 13, 15, 21, 23, 27, 29, 33, 39, 43, 45, 52, 56, 61, 67, 71, 78, 85, 90, 95, 102, 110, 117, 124, 131, 137, 145, 153, 163, 167, 180, 190, 196, 201, 211, 218, 233, 241, 252, 261, 271, 281, 290, 302, 314, 320, 329, 344, 355, 371, 385, 393, 407, 416, 423, 443

Offset

1,2

Comments

a(n) >= 1 for n >= 1 (Bertrand's postulate).

(2*n^2)/log(2*n^2) + (0.992*2*n^2)/log^2(2*n^2) - n^2/log(n^2) - (1.2762*n^2)/log^2(n^2) < a(n) < (2*n^2)/log(2*n^2) + (1.2762*2*n^2)/log^2(2*n^2) - n^2/log(n^2) - (0.992*n^2)/log^2(n^2) (for n^2 > 599 - see at LINKS for Dusart) holds for a(n) except n = {1,2,4,10}.

It seems a(n) is strictly monotonically increasing.

It seems that lim_{n->inf} a(n)/A(n) == 1, with A(n) = (2*n^2)/log(2*n^2) + (2*n^2)/log^2(2*n^2)) - n^2/log(n^2) - n^2/log(n^2).

Links

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

Pierre Dusart, The k-th prime is greater than k(ln k + ln ln k-1) for k>=2, Mathematics of Computation 68: (1999), 411-415.

Wikipedia, Bertrand's postulate

Formula

a(n) = pi(2n^2) - pi(n^2).

a(n) = A060715(n^2) for n > 1.

Maple

with(numtheory): A290564:=n->pi(2*n^2)-pi(n^2): seq(A290564(n), n=1..100); # Wesley Ivan Hurt, Aug 06 2017

Mathematica

Table[PrimePi[2 n^2] - PrimePi[n^2], {n, 1, 100}]

Prog

(PARI) a(n) = primepi(2*n^2) - primepi(n^2); \\ Michel Marcus, Aug 06 2017

Crossrefs

Cf. A000720, A060715.

Sequence in context: A341158 A294849 A320226 * A167803 A367402 A092213

Adjacent sequences: A290561 A290562 A290563 * A290565 A290566 A290567

Keyword

nonn,easy

Author

Ralf Steiner, Aug 06 2017

Status

approved