A160380 -id:A160380 - cp's OEIS frontend

A160383 Number of 3's in base-4 representation of n.

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 0

Frank Ruskey, Jun 05 2009

F. T. Adams-Watters and F. Ruskey, Generating Functions for the Digital Sum and Other Digit Counting Sequences, JIS 12 (2009) 09.5.6.

Cf. A007090 (base 4), A160380 (0's), A160381 (1's), A160382 (2's).

Cf. A283316 (mod 2).

a(n) = #select(x->(x==3), digits(n, 4)); \\ Michel Marcus, Mar 24 2020

Recurrence relation: a(0) = 0, a(4m+3) = 1+a(m), a(4m) = a(4m+1) = a(4m+2) = a(m).
G.f.: (1/(1-z))*Sum_{m>=0} (z^(3*4^m)*(1 - z^(4^m))/(1 - z^(4^(m+1)))).
Morphism: 0, j -> j,j,j,j+1; e.g., 0 -> 0001 -> 0001000100011112 -> ...

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

Original entry on oeis.org

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

Antti Karttunen, Sep 15 2017

			   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

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

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

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

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

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

(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)))))))

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

A031466 Numbers whose base-4 representation has one fewer 0 than 3's.

Original entry on oeis.org

3, 7, 11, 13, 14, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 79, 87, 91, 93, 94, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205

Offset: 1

Clark Kimberling

Numbers n such that A160383(n) - A160380(n) = 1. - Robert Israel, Jun 05 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A007090, A160380, A160383.

filter:= proc(n) local L;
  L:= convert(n,base,4);
  numboccur(3,L) - numboccur(0,L)=1
end proc:
select(filter, [$1..300]); # Robert Israel, Jun 05 2018

Select[Range[210],DigitCount[#,4,0]==DigitCount[#,4,3]-1&] (* Harvey P. Dale, Dec 16 2011 *)

A039003 Numbers whose base-4 representation has the same number of 0's and 2's.

Original entry on oeis.org

1, 3, 5, 7, 8, 13, 15, 18, 21, 23, 24, 29, 31, 33, 35, 36, 44, 50, 53, 55, 56, 61, 63, 70, 73, 75, 78, 82, 85, 87, 88, 93, 95, 97, 99, 100, 108, 114, 117, 119, 120, 125, 127, 130, 133, 135, 136, 141, 143, 145, 147, 148, 156, 160, 177, 179, 180, 188, 198, 201, 203

Offset: 1

Olivier Gérard

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A007090, A160380, A160382.

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A160383 Number of 3's in base-4 representation of n.

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

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A031466 Numbers whose base-4 representation has one fewer 0 than 3's.

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A039003 Numbers whose base-4 representation has the same number of 0's and 2's.

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