A074938 Odd numbers such that base 3 representation contains no 2.

1, 3, 9, 13, 27, 31, 37, 39, 81, 85, 91, 93, 109, 111, 117, 121, 243, 247, 253, 255, 271, 273, 279, 283, 325, 327, 333, 337, 351, 355, 361, 363, 729, 733, 739, 741, 757, 759, 765, 769, 811, 813, 819, 823, 837, 841, 847, 849, 973, 975, 981, 985, 999, 1003, 1009

Offset: 0

Author

Benoit Cloitre, Oct 04 2002; Nov 15 2003

Keywords

• easy
• nonn

Comments

Odd numbers in A005836.

Numbers m such that coefficient of x^m equals -1 in Product_{k>=0} 1-x^(3^k).

Numbers k such that binomial(2k, k) == 2 (mod 3).

Sum of an odd number of distinct powers of 3. - Emeric Deutsch, Dec 03 2003

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10000
• Emeric Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, arXiv:math/0407326 [math.CO], 2004; J. Num. Theory 117 (2006), 191-215.

Crossrefs

Intersection of A005408 and A005836.

Cf. A006996, A074939.

Programs

Mathematica

Select[Range[1,1111,2],Count[IntegerDigits[#,3],2]==0&] (* Harvey P. Dale, Dec 19 2010 *)

Python

def A074938(n): return int(bin((n<<1)+(n.bit_count()&1^1))[2:],3) # Chai Wah Wu, Jun 26 2025

Formula

a(n) (mod 3) = A010059(n).

((a(n)-1)/2) (mod 3) = A010060(n) = (1/2)*{binomial(2*a(n)+1, a(n)) (mod 3)}.