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A225858 Numbers of the form 2^i*3^j*(12k+7) or 2^i*3^j*(12k+11), i, j, k >= 0.

%I A225858 #29 Feb 24 2025 15:03:56
%S A225858 7,11,14,19,21,22,23,28,31,33,35,38,42,43,44,46,47,55,56,57,59,62,63,
%T A225858 66,67,69,70,71,76,79,83,84,86,88,91,92,93,94,95,99,103,105,107,110,
%U A225858 112,114,115,118,119,124,126,127,129,131,132,134,138,139,140
%N A225858 Numbers of the form 2^i*3^j*(12k+7) or 2^i*3^j*(12k+11), i, j, k >= 0.
%C A225858 The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Nov 14 2023
%H A225858 Amiram Eldar, <a href="/A225858/b225858.txt">Table of n, a(n) for n = 1..10000</a>
%e A225858 From _David A. Corneth_, Nov 11 2023: (Start)
%e A225858 28 = 2^2 * 7 = 2^2 * 3^0 * (12*0 + 7) is in the sequence as it meets the first form.
%e A225858 76 = 2^2 * 19 = 2^2 * 3^0 * (12*1 + 7) is in the sequence as it meets the first form.
%e A225858 15 = 3 * 5 = 2^0 * 3^1 * (12*0 + 5) is not in the sequence as it does not match any of the desired forms. (End)
%t A225858 Select[Range[140], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 3 &] (* _Amiram Eldar_, Nov 14 2023 *)
%o A225858 (PARI) for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%4==3),print1(n,",")))
%o A225858 (Magma) [n: n in [1..200] | d mod 4 eq 3 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // _Bruno Berselli_, May 16 2013
%o A225858 (Python)
%o A225858 from itertools import count
%o A225858 from sympy import integer_log
%o A225858 def A225858(n):
%o A225858     def f(x):
%o A225858         c = n
%o A225858         for i in range(integer_log(x,3)[0]+1):
%o A225858             i2 = 3**i
%o A225858             for j in count(0):
%o A225858                 k = i2<<j
%o A225858                 if k>x:
%o A225858                     break
%o A225858                 m = x//k
%o A225858                 c += (m-1)//12+(m-5)//12+2
%o A225858         return c
%o A225858     m, k = n, f(n)
%o A225858     while m != k: m, k = k, f(k)
%o A225858     return m # _Chai Wah Wu_, Feb 24 2025
%Y A225858 Complement of A225857. Cf. A225838.
%K A225858 nonn,easy
%O A225858 1,1
%A A225858 _Ralf Stephan_, May 18 2013
%E A225858 Name clarified by _Peter Munn_, Nov 11 2023