cp's OEIS Frontend

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

%I A225838 #25 Feb 02 2025 19:32:38
%S A225838 5,10,11,15,17,20,22,23,29,30,33,34,35,40,41,44,45,46,47,51,53,58,59,
%T A225838 60,65,66,68,69,70,71,77,80,82,83,87,88,89,90,92,94,95,99,101,102,105,
%U A225838 106,107,113,116,118,119,120,123,125,130,131,132,135,136,137,138
%N A225838 Numbers of form 2^i*3^j*(6k+5), i, j, k >= 0.
%C A225838 Are a(n) > A225837(n) for all n? - _Zak Seidov_, May 17 2013
%C A225838 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
%C A225838 The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Apr 03 2022
%H A225838 Zak Seidov, <a href="/A225838/b225838.txt">Table of n, a(n) for n = 1..10000</a>
%H A225838 Zak Seidov, <a href="/A225838/a225838.jpg">Graph of A225838(n) - A225837(n) for n=1..126454.</a>
%t A225838 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 *)
%o A225838 (PARI) for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%6==5),print1(n,",")))
%o A225838 (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
%o A225838 (Python)
%o A225838 from sympy import integer_log
%o A225838 def A225838(n):
%o A225838     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()))
%o A225838     m, k = n, f(n)
%o A225838     while m != k: m, k = k, f(k)
%o A225838     return m # _Chai Wah Wu_, Feb 02 2025
%Y A225838 Complement of A225837.
%Y A225838 Symmetric difference of A003159 and A026225; of A189716 and A325424.
%K A225838 nonn,easy
%O A225838 1,1
%A A225838 _Ralf Stephan_, May 16 2013