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A096304 Numbers k such that 3k does not divide (6k-4)!/((3k-2)!*(3k-1)!).

%I A096304 #32 Jul 30 2025 00:57:47
%S A096304 1,2,3,4,5,9,10,11,12,13,14,27,28,29,30,31,32,36,37,38,39,40,41,81,82,
%T A096304 83,84,85,86,90,91,92,93,94,95,108,109,110,111,112,113,117,118,119,
%U A096304 120,121,122,243,244,245,246,247,248,252,253,254,255,256,257,270,271
%N A096304 Numbers k such that 3k does not divide (6k-4)!/((3k-2)!*(3k-1)!).
%C A096304 Equivalently, members of A019469 divisible by 3, divided by 3.
%C A096304 _Ralf Stephan_'s formula is that terms k written in ternary have an arbitrary least significant digit and above that only 0's and 1's (per A340051). - _Kevin Ryde_, May 22 2021
%C A096304 {3a(n)-1:n>=1} is the set of positive integers k such that the k-th central binomial coefficient is not divisible by (k+1)*(2k-1). Such integers k are characterized by the following property: k is congruent to 2 (mod 3), and at least one of k-1, k+1 has no 2's in its base-3 expansion. - _Valerio De Angelis_, Aug 08 2022
%H A096304 Harvey P. Dale, <a href="/A096304/b096304.txt">Table of n, a(n) for n = 1..600</a>
%H A096304 Kevin Ryde, <a href="/A096304/a096304.txt">Proof of Ralf Stephan's formula</a>
%H A096304 Math Stackexchange, <a href="https://math.stackexchange.com/questions/4494256/factors-of-central-binomial-coefficient/4501152#4501152">Factors of central binomial coefficient</a>
%H A096304 <a href="/index/Ar#3-automatic">Index entries for 3-automatic sequences</a>.
%F A096304 a(n) = 9 * A005836(floor(n/6)) + (n mod 6) (conjectured) (confirmed, see links).
%F A096304 G.f.: x*(1+2*x)/(1-x^3) + 3/(1-x) * Sum_{i>=0} 3^i * x^(3*2^i) / (1 + x^(3*2^i)). - _Kevin Ryde_, May 22 2021
%t A096304 Select[Range[300],Mod[(6#-4)!/((3#-2)!(3#-1)!),3#]!=0&] (* _Harvey P. Dale_, Jun 11 2019 *)
%o A096304 (PARI) for(n=1,300,if(((6*n-4)!/(3*n-2)!/(3*n-1)!)%(3*n),print1(n",")))
%o A096304 (PARI) a(n) = my(r);[n,r]=divrem(n,3); fromdigits(concat(binary(n),r), 3); \\ _Kevin Ryde_, May 22 2021
%o A096304 (Python)
%o A096304 def A096304(n):
%o A096304     a, b = divmod(n,3)
%o A096304     return int(bin(a)[2:],3)*3+b # _Chai Wah Wu_, Jul 29 2025
%Y A096304 Cf. A340051 (ternary digits), A005836, A019469, A187358.
%K A096304 nonn,easy
%O A096304 1,2
%A A096304 _Ralf Stephan_, Aug 03 2004