A343598 -id:A343598 - cp's OEIS frontend

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

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

A343597 Numbers divisible by a 7-smooth composite number.

Original entry on oeis.org

4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130

Offset: 1

Peter Munn, Apr 21 2021

Numbers divisible by at least one of 4, 6, 9, 10, 14, 15, 21, 25, 35, 49.
Exactly half of the first 10, first 100 and first 600 positive integers are divisible by a 7-smooth composite number; the largest 7-smooth divisor of the remaining numbers is 1, 2, 3, 5 or 7.
Intervals extending to hundreds of integers with exactly 50% membership of this sequence are far from rare, some notable examples being [3000, 3999], [8000, 8999], [20000, 20999], [21000, 21999] and [23000, 23999]. This reflects the asymptotic density of the corresponding set being close to 0.5, precisely 1847 / 3675 = 0.50258503... (and membership of the set has a periodic pattern). See A343598 for further information.

			33 = 11 * 3 has divisors 1, 3, 11, 33, of which only 33 is composite. 33 is not 7-smooth, as its prime factors include 11, which is greater than 7. So 33 is not in the sequence.
52 = 13 * 2 * 2 is divisible by 4, which is composite and 7-smooth, so 52 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Smooth Number.

Cf. A002473 (7-smooth numbers), A014673, A020639, A210615 (smallest smoothest composite divisor), A343598.

Subsequence of A080672.

Select[Range[130], Plus @@ IntegerExponent[#, {2, 3, 5, 7}] > 1 &] (* Amiram Eldar, May 04 2021 *)

{a(n)} = {k : k >= 1, 2 <= A014673(k) <= 7}, where A014673(k) = lpf(k/lpf(k)), where lpf(m) = A020639(m), the least prime factor of m.
For n >= 1, a(22164 + n) = 44100 + a(n).
For n < 22164, a(22164 - n) = 44100 - a(n).

cp's OEIS Frontend

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

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