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
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
KEYWORD
nonn,easy
AUTHOR
Ralf Steiner, Aug 06 2017
STATUS
approved