A382746 If k appears, 8*k does not.

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Author

Jan Snellman, Apr 04 2025

Keywords

• nonn
• easy

Comments

Also: integers of the form 2^m*r, r odd, m congruent to 0, 1, 2 mod 6.

The asymptotic density of this sequence is 8/9. - Amiram Eldar, May 31 2025

Examples

			8, 16, ... , 56 are removed, but 8*8 = 64 remains.
		

Links

• Jan Snellman, Table of n, a(n) for n = 1..8889
• Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Crossrefs

Cf. A003159, A007417, A382744, A382745.

Programs

Mathematica

Select[Range[100], Mod[IntegerExponent[#, 2], 6] < 3 &] (* Amiram Eldar, Apr 04 2025 *)

PARI

isok(k) = valuation(k, 2) % 6 < 3; \\ Amiram Eldar, May 31 2025

Python

from functools import lru_cache
@lru_cache(maxsize=None, typed=True)
def in_sieve(n, S):
    if n == 1:
        return True
    elif n in S:
        return False
    else:
        L = [s for s in S if (n % s) == 0]
        return all(not in_sieve(n/ell, S) for ell in L )
def nth_in_sieve(n, S):
    if n == 1:
        return 1
    else:
        i, m = 1, 1
        while i < n:
            m = m+1
            if in_sieve(m, S):
                i = i+1
    return m
def a(n):
    return nth_in_sieve(n, tuple([8]))

Python

def A382746(n):
    def f(x): return n+x-sum((x>>m)+1>>1 for m in range(x.bit_length()+1) if m%6<3)
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

Sage

[ for  in range(1,100) if (valuation(_,2) % 6) < 3]