A079715 a(n) = pi(n) - pi(sqrt(n)) + 1.

1, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19

Offset: 1

Author

Benoit Cloitre, Feb 16 2003

Keywords

• nonn

Comments

A well-known application of the principle of inclusion-exclusion used in sieve methods.

Number of numbers less than or equal to n and coprime to the product of the primes less than sqrt(n), i.e., to A104588(n). - Lekraj Beedassy, Mar 17 2005

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000720, A056811, A056812, A104588.

Programs

Mathematica

Table[PrimePi[n] - PrimePi[Sqrt[n]] + 1, {n, 1, 100}] (* G. C. Greubel, May 13 2017 *)

PARI

for(n=1,100, print1(primepi(n) - primepi(sqrt(n)) + 1, ", ")) \\ G. C. Greubel, May 13 2017

Formula

a(n) = pi(n) - pi(sqrt(n)) + 1 = A000720(n) - A056811(n) + 1 = A056812(n) + 1.

a(n) = Sum_{k=1..n} mu(k)*floor(n/k) where each prime factor of k is <= sqrt(n). [Corrected by Steven Foster Clark, May 03 2023]

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 12 2007