A160382 -id:A160382 - cp's OEIS frontend

A292372 A binary encoding of 2-digits in base-4 representation of n.

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

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        1          2           1
   3        0          3           0
   4        0         10           0
   5        0         11           0
   6        1         12           1
   7        0         13           0
   8        2         20          10
   9        2         21          10
  10        3         22          11
  11        2         23          10
  12        0         30           0
  13        0         31           0
  14        1         32           1
  15        0         33           0
  16        0        100           0
  17        0        101           0
  18        1        102           1

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

Cf. A004198, A007088, A007090, A048724, A059906, A160382, A292370, A292371, A292373.

Cf. A289814 (analogous sequence for base-3).

Table[FromDigits[IntegerDigits[n, 4] /. k_ /; IntegerQ@ k :> If[k == 2, 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==2 else '0' for i in k), 2)
print([a(n) for n in range(121)]) # Indranil Ghosh, Sep 21 2017

def A292372(n): return 0 if (m:=n&~(n<<1)) < 2 else int(bin(m)[-2:1:-2][::-1],2) # Chai Wah Wu, Jun 30 2022

a(n) = A059906(n AND A048724(n)), where AND is a bitwise-AND (A004198).

For all n >= 0, A000120(a(n)) = A160382(n).

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

Original entry on oeis.org

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 -> ...

A329101 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the base 4 expansion of n equals the number of 2's in the base 4 expansion of a(n).

Original entry on oeis.org

0, 2, 1, 3, 6, 10, 8, 9, 4, 11, 5, 7, 12, 14, 13, 15, 18, 26, 22, 24, 34, 42, 38, 40, 25, 41, 27, 30, 32, 43, 33, 35, 16, 36, 17, 19, 37, 46, 39, 44, 20, 45, 21, 23, 28, 47, 29, 31, 48, 50, 49, 51, 54, 58, 56, 57, 52, 59, 53, 55, 60, 62, 61, 63, 66, 74, 70, 72

Offset: 0

Rémy Sigrist, Nov 07 2019

This sequence is a permutation of the nonnegative integers with inverse A329180.
Apparently, fixed points correspond to A001196.
The sequence has fractal features; for any k >= 0, the set of points { (n, a(n)), n = 0..4^k-1 } is symmetrical relative to the line of equation y + x = 4^k - 1 (see scatterplots in Links section).

			The first terms, alongside the base 4 representations of n and of a(n), are:
  n   a(n)  qua(n)  qua(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     2       1          2
   2     1       2          1
   3     3       3          3
   4     6      10         12
   5    10      11         22
   6     8      12         20
   7     9      13         21
   8     4      20         10
   9    11      21         23
  10     5      22         11
  11     7      23         13
  12    12      30         30
  13    14      31         32
  14    13      32         31
  15    15      33         33

Rémy Sigrist, Table of n, a(n) for n = 0..4095
Rémy Sigrist, PARI program for A329101
Rémy Sigrist, Scatterplot of the sequence for n = 0..4^3-1
Rémy Sigrist, Scatterplot of the sequence for n = 0..4^10-1
Rémy Sigrist, Colored scatterplot of the sequence for n = 0..4^10-1 (where the color is function of A160381(n))
Index entries for sequences that are permutations of the natural numbers

Cf. A001196, A160381, A160382, A329180.

PARI
```
See Links section.
```

A160381(n) = A160382(a(n)).

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.

A039006 Numbers whose base-4 representation has the same number of 2's and 3's.

Original entry on oeis.org

0, 1, 4, 5, 11, 14, 16, 17, 20, 21, 27, 30, 35, 39, 44, 45, 50, 54, 56, 57, 64, 65, 68, 69, 75, 78, 80, 81, 84, 85, 91, 94, 99, 103, 108, 109, 114, 118, 120, 121, 131, 135, 140, 141, 147, 151, 156, 157, 175, 176, 177, 180, 181, 187, 190, 194, 198, 200, 201, 210

Offset: 1

Olivier Gérard

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

Cf. A007090, A160382, A160383, A039005.

Cf. A000302, A052539 (subsequences).

Select[Range[0,250],DigitCount[#,4,2]==DigitCount[#,4,3]&] (* Harvey P. Dale, Mar 19 2017 *)

A338854 Product of the nonzero digits of (n written in base 4).

Original entry on oeis.org

1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 2, 2, 4, 6, 2, 2, 4, 6, 4, 4, 8, 12, 6, 6, 12, 18, 3, 3, 6, 9, 3, 3, 6, 9, 6, 6, 12, 18, 9, 9, 18, 27, 1, 1, 2, 3, 1, 1, 2, 3, 2, 2, 4, 6, 3, 3, 6, 9, 1

Offset: 0

Ilya Gutkovskiy, Nov 12 2020

Cf. A007090, A051801, A117592, A160382, A160383, A309954.

Table[Times @@ DeleteCases[IntegerDigits[n, 4], 0], {n, 0, 80}]
nmax = 80; A[] = 1; Do[A[x] = (1 + x + 2 x^2 + 3 x^3) A[x^4] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

a(n) = vecprod(select(x->x, digits(n, 4))); \\ Michel Marcus, Nov 12 2020

G.f. A(x) satisfies: A(x) = (1 + x + 2*x^2 + 3*x^3) * A(x^4).

a(n) = 2^A160382(n) * 3^A160383(n).

cp's OEIS Frontend

A292372 A binary encoding of 2-digits in base-4 representation of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Python

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A329101 Lexicographically earliest sequence of distinct nonnegative integers such that for any n >= 0, the number of 1's in the base 4 expansion of n equals the number of 2's in the base 4 expansion of a(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Links

Crossrefs

A039006 Numbers whose base-4 representation has the same number of 2's and 3's.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A338854 Product of the nonzero digits of (n written in base 4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula