A325424 Complement of A036668: numbers not of the form 2^i*3^j*k, i + j even, (k,6) = 1.

2, 3, 8, 10, 12, 14, 15, 18, 21, 22, 26, 27, 32, 33, 34, 38, 39, 40, 46, 48, 50, 51, 56, 57, 58, 60, 62, 69, 70, 72, 74, 75, 82, 84, 86, 87, 88, 90, 93, 94, 98, 104, 105, 106, 108, 110, 111, 118, 122, 123, 126, 128, 129, 130, 132, 134, 135, 136, 141, 142

Offset: 1

Author

Clark Kimberling, Apr 26 2019

Keywords

• nonn
• easy

Comments

These are the numbers 2x and 3x as x ranges through the numbers in A036668.

Numbers whose squarefree part is divisible by exactly one of {2, 3}. - Peter Munn, Aug 24 2020

The asymptotic density of this sequence is 5/12. - Amiram Eldar, Sep 20 2020

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Symmetric difference

Crossrefs

Cf. A325417, A036668.

Symmetric difference of: A003159 and A007417; A036554 and A145204\{0}.

Programs

Mathematica

a = {1}; Do[AppendTo[a, NestWhile[# + 1 &, Last[a] + 1, Apply[Or,
Map[MemberQ[a, #] &, Select[Flatten[{#/3, #/2}],
IntegerQ]]] &]], {150}]; a     (* A036668 *)
Complement[Range[Last[a]], a]  (* A325424 *)
(* Peter J. C. Moses, Apr 23 2019 *)

Python

from itertools import count
def A325424(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c = n
        for i in range(x.bit_length()+1):
            i2 = 1<x:
                    break
                m = x//k
                c += (m-1)//6+(m-5)//6+2
        return c
    return bisection(f,n,n) # Chai Wah Wu, Jan 28 2025

Formula

(2 * {A036668}) union (3 * {A036668}). - Sean A. Irvine, May 19 2019