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

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, 83, 84, 85, 86, 90, 91, 92, 93, 94, 95, 108, 109, 110, 111, 112, 113, 117, 118, 119, 120, 121, 122, 243, 244, 245, 246, 247, 248, 252, 253, 254, 255, 256, 257, 270, 271

Offset: 1

Author

Ralf Stephan, Aug 03 2004

Keywords

• nonn
• easy

Comments

Equivalently, members of A019469 divisible by 3, divided by 3.

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

{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

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..600
• Kevin Ryde, Proof of Ralf Stephan's formula
• Math Stackexchange, Factors of central binomial coefficient
• Index entries for 3-automatic sequences.

Crossrefs

Cf. A340051 (ternary digits), A005836, A019469, A187358.

Programs

Mathematica

Select[Range[300],Mod[(6#-4)!/((3#-2)!(3#-1)!),3#]!=0&] (* Harvey P. Dale, Jun 11 2019 *)

PARI

for(n=1,300,if(((6*n-4)!/(3*n-2)!/(3*n-1)!)%(3*n),print1(n",")))

PARI

a(n) = my(r);[n,r]=divrem(n,3); fromdigits(concat(binary(n),r), 3); \\ Kevin Ryde, May 22 2021

Python

def A096304(n):
    a, b = divmod(n,3)
    return int(bin(a)[2:],3)*3+b # Chai Wah Wu, Jul 29 2025

Formula

a(n) = 9 * A005836(floor(n/6)) + (n mod 6) (conjectured) (confirmed, see links).

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