A370662 Numbers m such that 3 does not divide the m-th Catalan number A000108(m); m such that A067397(m) = 0.

0, 1, 2, 3, 4, 8, 9, 10, 11, 12, 13, 26, 27, 28, 29, 30, 31, 35, 36, 37, 38, 39, 40, 80, 81, 82, 83, 84, 85, 89, 90, 91, 92, 93, 94, 107, 108, 109, 110, 111, 112, 116, 117, 118, 119, 120, 121, 242, 243, 244, 245, 246, 247, 251, 252, 253, 254, 255, 256, 269, 270, 271, 272, 273, 274

Offset

1,3

Comments

A number m is a term if and only if m = 3*k-1 (k>0), 3*k or 3*k+1 for k in A005836; see A067397.

Conjecture: the only terms of the form 2^r-1 are 0, 1, 3, 31 and 255. Since 2^r-1 !== 2 (mod 3), this is equivalent to saying that the only numbers of the form 2^r-1 that have no digits 2 in ternary are 0, 1, 3, 31, 255. The conjecture would imply that the n-th Catalan number is divisible by 2 or 3 other than n taking these values.

Links

Jianing Song, Table of n, a(n) for n = 1..12287 (all terms < 3^13-1)

Formula

a(3*k-3) = 3*A005836(k)-1 (k>1), a(3*k-2) = 3*A005836(k), a(3*k-1) = 3*A005836(k)+1.

a(3*2^r-1) = (3^(r+1)-1)/2, a(3*2^r) = 3^(r+1)-1.

Example

11 is a term since that 3 does not divide the 11th Catalan number 58786. Note that 11 = 3*4 - 1, and 4 is in A005836.
31 is a term since that 3 does not divide the 31st Catalan number 14544636039226909. Note that 31 = 3*10 + 1, and 10 is in A005836.

Prog

(PARI) a(n) = 3*fromdigits(binary(n\3), 3) + n%3 - 1 \\ adapted from Gheorghe Coserea's program for A005836
(Python)
def A370662(n):
    a, b = divmod(n, 3)
    return 3*int(bin(a)[2:], 3)+b-1 # Chai Wah Wu, Feb 29 2024

Crossrefs

Cf. A000108, A067397, A000989, A005836.

Sequence in context: A172002 A274927 A066338 * A047453 A037467 A165564

Adjacent sequences: A370659 A370660 A370661 * A370663 A370664 A370665

Keyword

nonn,easy

Author

Jianing Song, Feb 24 2024

Status

approved