A073425 a(0)=0; for n>0, a(n) = number of primes not exceeding n-th composite number.

0, 2, 3, 4, 4, 4, 5, 6, 6, 6, 7, 8, 8, 8, 9, 9, 9, 9, 9, 10, 11, 11, 11, 11, 11, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 19, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 22, 23, 23, 23, 23, 23, 24, 24, 24, 24, 24, 24, 24, 25, 25, 25, 26

Offset: 0

Author

Labos Elemer, Jul 31 2002

Keywords

• nonn

Comments

a(n-1) = A018252(n) - n. a(n-1) = inverse (frequency distribution) sequence of A014689(n), i.e. number of terms of sequence A014689(n) less than n. a(n) = A073169(n+1) - 1, for n >= 1. For n >= 1: a(n) + 1 = A073169(n) = the number of set {1, primes}, i.e. (A008578) less than (n)-th composite numbers (A002828(n)). a(n-1) = The number of primes (A000040(n)) less than n-th nonprime (A018252(n)). - Jaroslav Krizek, Jun 27 2009

Examples

			n=100: composite[100]=133,Pi[133]=32=a(100)
		

Crossrefs

Cf. A065890, A073426, A000720, A002808, A000040, A018252, A158611, A073169.

Programs

Mathematica

c[x_] := FixedPoint[x+PrimePi[ # ]+1&, x] Table[PrimePi[c[w]], {w, 1, 128}]
With[{nn=150},PrimePi/@Complement[Range[nn],Prime[Range[PrimePi[nn]]]]] (* Harvey P. Dale, Jun 26 2013 *)

Python

from sympy import composite
def A073425(n): return composite(n)-n-1 if n else 0 # Chai Wah Wu, Oct 11 2024

Formula

a(n) = A000720(A002808(n)).

a(n) ~ n. - Charles R Greathouse IV, Sep 02 2015

a(n) = A002808(n)-n-1 for n > 0. - Chai Wah Wu, Oct 11 2024

Extensions

Edited by N. J. A. Sloane, Jul 04 2009 at the suggestion of R. J. Mathar

Correction for change of offset in A158611 and A008578 in Aug 2009 Jaroslav Krizek, Jan 27 2010