cp's OEIS Frontend

A225857 Numbers of the form 2^i*3^j*(12k+1) or 2^i*3^j*(12k+5), i, j, k >= 0.

%I A225857 #33 Feb 24 2025 14:39:58
%S A225857 1,2,3,4,5,6,8,9,10,12,13,15,16,17,18,20,24,25,26,27,29,30,32,34,36,
%T A225857 37,39,40,41,45,48,49,50,51,52,53,54,58,60,61,64,65,68,72,73,74,75,77,
%U A225857 78,80,81,82,85,87,89,90,96,97,98,100,101,102,104,106,108,109,111
%N A225857 Numbers of the form 2^i*3^j*(12k+1) or 2^i*3^j*(12k+5), i, j, k >= 0.
%C A225857 From _Peter Munn_, Nov 11 2023: (Start)
%C A225857 Numbers k whose 5-rough part, A065330(k), is congruent to 1 modulo 4.
%C A225857 Contains all nonzero squares.
%C A225857 Positive integers in the multiplicative subgroup of rationals generated by 2, 3, 5 and integers congruent to 1 modulo 12. Thus, the sequence is closed under multiplication and, provided the result is an integer, under division.
%C A225857 This subgroup has index 2 and does not include -1, so is the complement of its negation. In respect of the sequence, the index 2 property implies we can take any absent positive integer m, and divide by m all terms that are multiples of m to get the complementary sequence, A225858.
%C A225857 Likewise, the sequence forms a subgroup of index 2 of the positive integers under the operation A059897(.,.).
%C A225857 (End)
%C A225857 The asymptotic density of this sequence is 1/2. - _Amiram Eldar_, Nov 14 2023
%H A225857 Amiram Eldar, <a href="/A225857/b225857.txt">Table of n, a(n) for n = 1..10000</a>
%t A225857 Select[Range[120], Mod[#/Times @@ ({2, 3}^IntegerExponent[#, {2, 3}]), 4] == 1 &] (* _Amiram Eldar_, Nov 14 2023 *)
%o A225857 (PARI) for(n=1,200,t=n/(2^valuation(n,2)*3^valuation(n,3));if((t%4==1),print1(n,",")))
%o A225857 (Magma) [n: n in [1..200] | d mod 4 eq 1 where d is n div (2^Valuation(n,2)*3^Valuation(n,3))]; // _Bruno Berselli_, May 16 2013
%o A225857 (Python)
%o A225857 from itertools import count
%o A225857 from sympy import integer_log
%o A225857 def A225857(n):
%o A225857     def f(x):
%o A225857         c = n
%o A225857         for i in range(integer_log(x,3)[0]+1):
%o A225857             i2 = 3**i
%o A225857             for j in count(0):
%o A225857                 k = i2<<j
%o A225857                 if k>x:
%o A225857                     break
%o A225857                 m = x//k
%o A225857                 c += (m-7)//12+(m-11)//12+2
%o A225857         return c
%o A225857     m, k = n, f(n)
%o A225857     while m != k: m, k = k, f(k)
%o A225857     return m # _Chai Wah Wu_, Feb 24 2025
%Y A225857 Complement of A225858.
%Y A225857 Cf. A059897, A065330, A225837.
%K A225857 nonn,easy
%O A225857 1,2
%A A225857 _Ralf Stephan_, May 18 2013
%E A225857 Name clarified by _Peter Munn_, Nov 10 2023