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

1,2

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

Links

Jan Snellman, Table of n, a(n) for n = 1..8889

Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 24.

Example

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

Mathematica

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

Prog

(SageMath)
[_ for _ in range(1, 100) if (valuation(_, 2) % 6) < 3]
(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
(PARI) isok(k) = valuation(k, 2) % 6 < 3; \\ Amiram Eldar, May 31 2025

Crossrefs

Cf. A003159, A007417, A382744, A382745.

Sequence in context: A329135 A182768 A368917 * A207481 A047592 A187320

Adjacent sequences: A382743 A382744 A382745 * A382747 A382748 A382749

Keyword

nonn,easy

Author

Jan Snellman, Apr 04 2025

Status

approved