A292370 A binary encoding of the zeros in base-4 representation of n.

0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 7, 6, 6, 6, 5, 4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 3, 2, 2, 2, 1, 0, 0, 0, 1

Offset: 0

Author

Antti Karttunen, Sep 15 2017

Keywords

• nonn
• base

Examples

			   n      a(n)     base-4(n)  binary(a(n))
                  A007090(n)  A007088(a(n))
  --      ----    ----------  ------------
   1        0          1           0
   2        0          2           0
   3        0          3           0
   4        1         10           1
   5        0         11           0
   6        0         12           0
   7        0         13           0
   8        1         20           1
   9        0         21           0
  10        0         22           0
  11        0         23           0
  12        1         30           1
  13        0         31           0
  14        0         32           0
  15        0         33           0
  16        3        100          11
  17        2        101          10
		

Links

• Antti Karttunen, Table of n, a(n) for n = 0..65536
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A007088, A007090, A160380, A292371, A292372, A292373.

Cf. A291770 (analogous sequence for base-3).

Programs

Mathematica

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 0, 1, 0], 2], {n, 0, 120}] (* Michael De Vlieger, Sep 21 2017 *)

Python

from sympy.ntheory.factor_ import digits
def a(n):
    k=digits(n, 4)[1:]
    return 0 if n==0 else int("".join('1' if i==0 else '0' for i in k), 2)
print([a(n) for n in range(111)]) # Indranil Ghosh, Sep 21 2017

Scheme

(define (A292370 n) (if (zero? n) n (let loop ((n n) (b 1) (s 0)) (if (< n 4) s (let ((d (modulo n 4))) (if (zero? d) (loop (/ n 4) (+ b b) (+ s b)) (loop (/ (- n d) 4) (+ b b) s)))))))

Formula

For all n >= 0, A000120(a(n)) = A160380(n).