A291770 A binary encoding of the zeros in ternary representation of n.

0, 0, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 7, 6, 6, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0, 0, 1, 0, 0, 3, 2, 2, 1, 0, 0, 1, 0, 0, 15, 14, 14, 13, 12, 12, 13, 12, 12, 11, 10, 10, 9, 8, 8, 9, 8, 8, 11, 10, 10, 9, 8, 8, 9, 8, 8, 7, 6, 6

Offset: 1

Author

Antti Karttunen, Sep 11 2017

Keywords

• nonn
• base

Comments

The ones in the binary representation of a(n) correspond to the nonleading zeros in the ternary representation of n. For example: ternary(33) = 1020 and binary(a(33)) = 101 (a(33) = 5).

Examples

			   n      a(n)    ternary(n)  binary(a(n))
                  A007089(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0            1           0
   2        0            2           0
   3        1           10           1
   4        0           11           0
   5        0           12           0
   6        1           20           1
   7        0           21           0
   8        0           22           0
   9        3          100          11
  10        2          101          10
  11        2          102          10
  12        1          110           1
  13        0          111           0
  14        0          112           0
  15        1          120           1
  16        0          121           0
  17        0          122           0
  18        3          200          11
  19        2          201          10
  20        2          202          10
  21        1          210           1
  22        0          211           0
  23        0          212           0
  24        1          220           1
  25        0          221           0
  26        0          222           0
  27        7         1000         111
  28        6         1001         110
  29        6         1002         110
  30        5         1010         101
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..59049

Crossrefs

Cf. A007088, A007089, A077267, A289813, A289814, A291771.

Programs

Mathematica

Table[FromDigits[IntegerDigits[n, 3] /. k_ /; k < 3 :> If[k == 0, 1, 0], 2], {n, 110}] (* Michael De Vlieger, Sep 11 2017 *)

Python

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 3)[1:]
    return int("".join('1' if i==0 else '0' for i in k), 2)
print([a(n) for n in range(1, 111)]) # Indranil Ghosh, Sep 21 2017

Scheme

(define (A291770 n) (if (zero? n) n (let loop ((n n) (b 1) (s 0)) (if (< n 3) s (let ((d (modulo n 3))) (if (zero? d) (loop (/ n 3) (+ b b) (+ s b)) (loop (/ (- n d) 3) (+ b b) s)))))))

Formula

For all n >= 0, a(A000244(n)) = A000225(n), that is, a(3^n) = (2^n) - 1. [The records in the sequence].

For all n >= 1, A000120(a(n)) = A077267(n).

For all n >= 1, A278222(a(n)) = A291771(n).