A080786 -id:A080786 - cp's OEIS frontend

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

N. J. A. Sloane

nonn

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

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

Cf. A000720, A080339, A147693.

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

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

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

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

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

A071604 a(n) is the number of 7-smooth numbers <= n.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 11, 11, 12, 13, 14, 14, 15, 15, 16, 17, 17, 17, 18, 19, 19, 20, 21, 21, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 34, 35, 36, 36, 36, 36, 36, 36, 37, 37, 38

Offset: 1

Benoit Cloitre, Jun 02 2002

A 7-smooth number is a number of the form 2^x*3^y*5^z*7^u, (x,y,z,u) >= 0.

In other words, a 7-smooth number is a number with no prime factor greater than 7. - Peter Munn, Nov 20 2021

			a(11) = 10 as there are 10 7-smooth numbers <= 11. Namely 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. - _David A. Corneth_, Apr 19 2021

David A. Corneth, Table of n, a(n) for n = 1..10000

Partial sums of A086299.
Column 7 of A080786.
Cf. A002473, A106600, A343598.
Equivalent sequences with other limits on greatest prime factor: A070939 (2), A071521 (3), A071520 (5), A071523 (11), A080684 (13), A080685 (17), A080686 (19), A096300 (log n).

for(n=1,100,print1(sum(k=1,n,if(sum(i=5,n,if(k%prime(i),0,1)),0,1)),","))

from sympy import integer_log
def A071604(n):
    c = 0
    for i in range(integer_log(n,7)[0]+1):
        i7 = 7**i
        m = n//i7
        for j in range(integer_log(m,5)[0]+1):
            j5 = 5**j
            r = m//j5
            for k in range(integer_log(r,3)[0]+1):
                c += (r//3**k).bit_length()
    return c # Chai Wah Wu, Sep 16 2024

a(n) = Card{ k | A002473 (k) <= n }.

Name corrected by David A. Corneth, Apr 19 2021

cp's OEIS Frontend

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

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