A382744 -id:A382744 - cp's OEIS frontend

A382745 If k appears, 7*k does not.

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 15, 16, 17, 18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 30, 31, 32, 33, 34, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 71, 72, 73, 74, 75, 76, 78, 79, 80, 81, 82, 83, 85

Offset: 1

Jan Snellman, Apr 04 2025

Also numbers with an even number of 7's in their prime factorization.

Natural density 7/8.

			7 is removed since 7 = 7*1, 14, 21, 28, 35, 42 are removed, but 49 remains.

Jan Snellman, Table of n, a(n) for n = 1..8751
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382744, A382746.

q:= n-> is(irem(padic[ordp](n,7), 2)=0):
select(q, [$1..85])[];  # Alois P. Heinz, Apr 04 2025

Select[Range[100], EvenQ[IntegerExponent[#, 7]] &] (* Amiram Eldar, Apr 04 2025 *)

def ok(n):
    c = 0
    while n and n%7 == 0: n //= 7; c += 1
    return c&1 == 0
print([k for k in range(1, 86) if ok(k)]) # Michael S. Branicky, Apr 04 2025

from sympy import integer_log
def A382745(n):
    def f(x): return n+x-sum((k:=x//7**m)-k//7 for m in range(0,integer_log(x,7)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Apr 10 2025

a(n) ~ (8/7)*n.

A382746 If k appears, 8*k does not.

Original entry on oeis.org

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

Jan Snellman, Apr 04 2025

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

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

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

Cf. A003159, A007417, A382744, A382745.

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

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

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]))

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

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

A382750 If k appears, 9*k does not.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81

Offset: 1

Jan Snellman, May 09 2025

Integers k with val(k, 9) even, where val(k, 9) is the 9-adic valuation of k.
Natural density 9/10.
Differs from A168183: 81 for example is not in A168183 but in this sequence. - R. J. Mathar, May 26 2025

			18 = 9*2 is not a term because 2 is a term.
162 = 9*18 is a term since 18 is not a term.

Jan Snellman, Table of n, a(n) for n = 1..9000
Jan Snellman, Greedy Regular Convolutions, arXiv:2504.02795 [math.NT], 2025.

Cf. A003159, A007417, A382744, A382745, A382746.

Select[Range[100], Mod[IntegerExponent[#, 3], 4] < 2 &] (* Amiram Eldar, May 12 2025 *)

from sympy import integer_log
def A382750(n):
    def f(x): return n+x-sum((k:=x//9**m)-k//9 for m in range(0,integer_log(x,9)[0]+1,2))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, May 24 2025

[ for  in range(1,100+1) if (valuation(_,3) % 4) < 2 ]

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A382745 If k appears, 7*k does not.

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A382746 If k appears, 8*k does not.

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A382750 If k appears, 9*k does not.

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