A036234 Number of primes <= n, if 1 is counted as a prime.

1, 2, 3, 3, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20, 20

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

This sequence is the largest nondecreasing sequence a(n) such that a(Prime(n)-1) = n. - Tanya Khovanova, Jun 20 2007

Partial sums of A080339. - Jaroslav Krizek, Mar 23 2009

Let G(n) be the graph whose vertices represent integers 1 through n, and where vertices a and b are adjacent iff gcd(a,b)>1. Then a(n) is the independence number of G(n). - Aaron Dunigan AtLee, May 23 2009

a(1)=1; a(n)= max[A061395(n), A061395(n-1)]. - Jacques ALARDET, Dec 28 2011

It appears that a(n) is the minimal index i for which binomial(k*prime(i), prime(i)) mod prime(i) = k. For example, binomial(11*prime(n), prime(n)) mod prime(n) produces the sequence 1,2,1,4,0,11,11,11,11 and a(11)=6. It also appears that binomial(k*prime(a(n)-1), prime(a(n)-1)) mod prime(a(n)-1) = 0 iff k is prime. - Gary Detlefs, Aug 05 2013

a(n) is the number of noncomposite numbers <= n. The noncomposite number are in A008578. - Omar E. Pol, Aug 31 2013

Number of distinct terms in n-th row of the triangle in A080786. - Reinhard Zumkeller, Sep 10 2013

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A080339, A147693.

Programs

Haskell

a036234 = (+ 1) . a000720  -- Reinhard Zumkeller, Sep 10 2013

Maple

A036234 := proc(n)
    if n = 1 then
        1;
    else
        1+numtheory[pi](n) ;
    end if;
end proc: # R. J. Mathar, Jan 28 2014

Mathematica

Table[PrimePi[n] + 1, {n, 100}] (* Tanya Khovanova, Jun 20 2007 *)

PARI

a(n)=primepi(n)+1 \\ Charles R Greathouse IV, Apr 29 2015

Formula

a(n) = A000720(n) + 1. - Jaroslav Krizek, Mar 23 2009