A225838 Numbers of form 2^i*3^j*(6k+5), i, j, k >= 0.

5, 10, 11, 15, 17, 20, 22, 23, 29, 30, 33, 34, 35, 40, 41, 44, 45, 46, 47, 51, 53, 58, 59, 60, 65, 66, 68, 69, 70, 71, 77, 80, 82, 83, 87, 88, 89, 90, 92, 94, 95, 99, 101, 102, 105, 106, 107, 113, 116, 118, 119, 120, 123, 125, 130, 131, 132, 135, 136, 137, 138

Offset: 1

Author

Ralf Stephan, May 16 2013

Keywords

• nonn
• easy

Comments

Are a(n) > A225837(n) for all n? - Zak Seidov, May 17 2013

Yes. Imagine every 3-smooth number, m, visits you regularly, depositing a gold coin for safe keeping at each epoch (6k+1)*m and collecting it at epoch (6k+5)*m. If you run out of coins, you are doing something other than keeping them in a vault! - Peter Munn, Nov 13 2023

The asymptotic density of this sequence is 1/2. - Amiram Eldar, Apr 03 2022

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000
• Zak Seidov, Graph of A225838(n) - A225837(n) for n=1..126454.

Crossrefs

Complement of A225837.

Symmetric difference of A003159 and A026225; of A189716 and A325424.

Programs

Magma

[n: n in [1..200] | d mod 6 eq 5 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // Bruno Berselli, May 16 2013

Mathematica

mx = 153; t = {}; Do[n = 2^i*3^j (6 k + 5); If[n <= mx, AppendTo[t, n]], {i, 0, Log[2, mx]}, {j, 0, Log[3, mx]}, {k, 0, mx/6}]; Union[t] (* T. D. Noe, May 16 2013 *)

PARI

for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%6==5),print1(n,",")))

Python

from sympy import integer_log
def A225838(n):
    def f(x): return n+sum(((x//3**i>>j)+5)//6 for i in range(integer_log(x,3)[0]+1) for j in range((x//3**i).bit_length()))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 02 2025