A291539 a(n) = PrimePi(n^3) - PrimePi(n) * PrimePi(n^2), where PrimePi = A000720.

0, 2, 1, 6, 3, 14, 8, 25, 41, 68, 67, 99, 93, 136, 188, 240, 229, 303, 306, 383, 467, 562, 566, 688, 795, 922, 1066, 1227, 1247, 1421, 1446, 1620, 1826, 2036, 2283, 2511, 2566, 2843, 3115, 3401, 3431, 3746, 3827, 4163, 4526, 4895, 4981, 5369, 5743, 6229, 6712, 7165, 7202, 7743, 8258, 8835, 9453, 9999, 10132, 10736

Offset: 1

Author

Jonathan Sondow, 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) * PrimePi(n)^2 for n > 1.

For PrimePi(n) * PrimePi(n^2) - PrimePi(n)^3, see A291540.

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

For prime(n) * prime(n^2) - prime(n^3), see A291541.

Examples

			a(2) = PrimePi(2^3) - PrimePi(2) * PrimePi(2^2) = 4 - 1 * 2 = 2.
		

Crossrefs

Cf. A000720, A123914, A262199, A291440, A291538, A291540, A291541, A291542.

Programs

Mathematica

Table[ PrimePi[n^3] - PrimePi[n]*PrimePi[n^2], {n, 60}]

PARI

a(n) = primepi(n^3) - primepi(n) * primepi(n^2); \\ Michel Marcus, Sep 10 2017

Formula

a(n) = A000720(n^3) - A000720(n) * A000720(n)^2.

a(n) = A291538(n) - A291540(n).

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