A291538 a(n) = PrimePi(n^3) - PrimePi(n)^3, where PrimePi = A000720.

0, 3, 1, 10, 3, 20, 4, 33, 65, 104, 92, 144, 111, 184, 260, 348, 313, 422, 370, 495, 635, 786, 728, 904, 1092, 1291, 1498, 1731, 1707, 1961, 1897, 2181, 2486, 2806, 3152, 3490, 3466, 3851, 4267, 4685, 4653, 5111, 5045, 5549, 6066, 6617, 6541, 7124, 7723, 8359, 9007, 9685, 9650, 10383, 11106, 11859, 12669, 13487, 13498, 14374

Offset: 1

Author

Jonathan Sondow, following a suggestion from Altug Alkan, Aug 25 2017

Keywords

• nonn

Comments

All terms are positive except a(1) = 0, by the PNT with error term for large n and computation for smaller n. In particular, PrimePi(n^3) > PrimePi(n)^3 for n > 1. Indeed, by A291539 and A291540, PrimePi(n^3) > PrimePi(n) * PrimePi(n^2) > PrimePi(n)^3 for n > 7.

For prime(n)^3 - prime(n^3), see A262199.

For PrimePi(n^2) - PrimePi(n)^2, see A291440.

Examples

			a(3) = PrimePi(3^3) - PrimePi(3)^3 = 9 - 2^3 = 1.
		

Crossrefs

Cf. A000720, A123914, A262199, A291440, A291539, A291540, A291541, A291542.

Programs

Mathematica

Table[ PrimePi[n^3] - PrimePi[n]^3, {n, 60}]

PARI

a(n) = primepi(n^3) - primepi(n)^3; \\ Michel Marcus, Sep 10 2017

Formula

a(n) = A000720(n^3) - A000720(n)^3.

a(n) = A291539(n) + A291540(n).

a(n) ~ (n^3 / log(n))*(1/3 - 1/log(n)^2) as n tends to infinity.