A291440 - OEIS

A291440

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

0, 1, 0, 2, 0, 2, -1, 2, 6, 9, 5, 9, 3, 8, 12, 18, 12, 17, 8, 14, 21, 28, 18, 24, 33, 41, 48, 56, 46, 54, 41, 51, 60, 70, 79, 89, 75, 84, 96, 107, 94, 105, 87, 99, 110, 123, 104, 117, 132, 142, 153, 168, 153, 165, 178, 189, 201, 218, 198, 214, 195, 208, 225, 240, 254, 270, 248, 263, 280, 293, 275, 290, 264, 281, 298, 316, 338, 352, 327, 350

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

COMMENTS

The only zero values are a(1) = a(3) = a(5) = 0. The only negative value is a(7) = -1. In particular, pi(n^2) > pi(n)^2 for n > 7. These can be proved by the PNT with error term for large n and computation for smaller n.

For prime(n)^2 - prime(n^2), see A123914.

For pi(n^3) - pi(n)^3, see A291538.

Mincu and Panaitopol (2008) prove that pi(m*n) >= pi(m)*pi(n) for all positive m and n except for m = 5, n = 7; m = 7, n = 5; and m = n = 7. This implies for m = n that a(n) >= 0 if n <> 7. - Jonathan Sondow, Nov 03 2017

Diagonal of the triangular array A294508. - Jonathan Sondow and Robert G. Wilson v, Nov 08 2017

LINKS

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

Gabriel Mincu and Laurentiu Panaitopol, Properties of some functions connected to prime numbers, J. Inequal. Pure Appl. Math., 9 No. 1 (2008), Art. 12.

FORMULA

a(n) = A000720(n^2) - A000720(n)^2.

a(n) ~ (n^2 / log(n))*(1/2 - 1/log(n)) as n tends to infinity, by the PNT.

From Jonathan Sondow and Robert G. Wilson v, Nov 08 2017: (Start)

a(n) = A294508(n*(n+1)/2).

a(n) >= A294509(n). (End)

EXAMPLE

a(7) = pi(7^2) - pi(7)^2 = 15 - 4^2 = -1.

MAPLE

seq(numtheory:-pi(n^2)-numtheory:-pi(n)^2, n=1..100); # Robert Israel, Aug 25 2017

MATHEMATICA

Table[PrimePi[n^2] - PrimePi[n]^2, {n, 80}]