A060373 -id:A060373 - cp's OEIS frontend

A060372 p(n), positive part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.

0, 1, 3, 3, 4, 9, 9, 10, 9, 9, 10, 12, 12, 13, 27, 27, 28, 27, 27, 28, 30, 30, 31, 27, 27, 28, 27, 27, 28, 30, 30, 31, 36, 36, 37, 36, 36, 37, 39, 39, 40, 81, 81, 82, 81, 81, 82, 84, 84, 85, 81, 81, 82, 81, 81, 82, 84, 84, 85, 90, 90, 91, 90, 90, 91, 93, 93, 94, 81, 81, 82, 81

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Apr 02 2001

The graphs of p(n), q(n) are fractals; the graph of p(n)+q(n) is Sierpiński-like.

			Example: 14=27-13=3^3 -(3^0+3^1+3^2), 16=28-12=3^3+3^0 -(3^1+3^2), 20=30-10=3^3+3^1 -(3^0+3^2); 27+13=28+12=30+10=40; 10,12,13, 27, 28, 30 are written without 2 in base 3.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A005836, A060373, A060374.

a060372 n = (a060374 n + n) `div` 2  -- Reinhard Zumkeller, Jun 09 2012

A343229 A binary encoding of the digits "-1" in balanced ternary representation of n.

Original entry on oeis.org

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, 5, 4, 4, 5, 4, 4, 3, 2, 2, 1, 0

Offset: 0

Rémy Sigrist, Apr 08 2021

The ones in the binary representation of a(n) correspond to the digits "-1" in the balanced ternary representation of n.

We can extend this sequence to negative indices: a(-n) = A343228(n) for any n >= 0.

			The first terms, alongside the balanced ternary representation of n (with "T" instead of digits "-1") and the binary representation of a(n), are:
  n   a(n)  ter(n)  bin(a(n))
  --  ----  ------  ---------
   0     0       0          0
   1     0       1          0
   2     1      1T          1
   3     0      10          0
   4     0      11          0
   5     3     1TT         11
   6     2     1T0         10
   7     2     1T1         10
   8     1     10T          1
   9     0     100          0
  10     0     101          0
  11     1     11T          1
  12     0     110          0
  13     0     111          0
  14     7    1TTT        111
  15     6    1TT0        110

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Wikipedia, Balanced ternary

Cf. A059095, A060373, A140267, A289814, A289831, A343228, A343230, A343231, A005836 (indices of 0's).

a(n) = { my (v=0, b=1, t); while (n, t=centerlift(Mod(n, 3)); if (t==-1, v+=b); n=(n-t)\3; b*=2); v }

a(n) = A289831(A060373(n)).

A060374 a(n)=p+q, where n=p-q and p, q, p+q are in A005836 (integers written without 2 in base 3).

Original entry on oeis.org

0, 1, 4, 3, 4, 13, 12, 13, 10, 9, 10, 13, 12, 13, 40, 39, 40, 37, 36, 37, 40, 39, 40, 31, 30, 31, 28, 27, 28, 31, 30, 31, 40, 39, 40, 37, 36, 37, 40, 39, 40, 121, 120, 121, 118, 117, 118, 121, 120, 121, 112, 111, 112, 109, 108, 109, 112, 111, 112, 121, 120, 121, 118

Offset: 0

Claude Lenormand (claude.lenormand(AT)free.fr), Apr 02 2001

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Zoran Sunic, Tree morphisms, transducers and integer sequences, arXiv:math/0612080 [math.CO], 2006.

Cf. A005836, A039966, A060372, A060373.

a060374 n = f $ dropWhile (< n) a005836_list where
   f (p:ps) | a039966 (p-n) == 1 && a039966 (2*p-n) == 1 = 2*p - n
            | otherwise                                  = f ps
-- Reinhard Zumkeller, Sep 29 2011

Definition clarified by Zoran Sunic, Feb 16 2006

A380180 Irregular table T(n, k), n >= 0, k = 1..2^A005812(n); the n-th row lists the integers m (possibly negative) such that the nonzero digits in the balanced ternary expansion of m appear in the balanced ternary expansion of n.

Original entry on oeis.org

0, 0, 1, -1, 0, 2, 3, 0, 3, 0, 1, 3, 4, -4, -3, -1, 0, 5, 6, 8, 9, -3, 0, 6, 9, -3, -2, 0, 1, 6, 7, 9, 10, -1, 0, 8, 9, 0, 9, 0, 1, 9, 10, -1, 0, 2, 3, 8, 9, 11, 12, 0, 3, 9, 12, 0, 1, 3, 4, 9, 10, 12, 13, -13, -12, -10, -9, -4, -3, -1, 0, 14, 15, 17, 18, 23, 24, 26, 27

Offset: 0

Rémy Sigrist, Jan 15 2025

Every integer appears infinitely many times in the sequence.

See A368239 (resp. A380181) for the nonnegative values (resp. the nonpositive values, negated) in order of appearance in the present sequence.

			Irregular table T(n, k) begins:
  n   n-th row
  --  -------------------------
   0  0
   1  0, 1
   2  -1, 0, 2, 3
   3  0, 3
   4  0, 1, 3, 4
   5  -4, -3, -1, 0, 5, 6, 8, 9
   6  -3, 0, 6, 9
   7  -3, -2, 0, 1, 6, 7, 9, 10
   8  -1, 0, 8, 9
   9  0, 9
  10  0, 1, 9, 10
  11  -1, 0, 2, 3, 8, 9, 11, 12
  12  0, 3, 9, 12
.
Irregular table T(n, k) begins in balanced ternary:
  n    n-th row
  ---  --------------------------------
    0  0
    1  0, 1
   1T  T, 0, 1T, 10
   10  0, 10
   11  0, 1, 10, 11
  1TT  TT, T0, T, 0, 1TT, 1T0, 10T, 100
  1T0  T0, 0, 1T0, 100
  1T1  T0, T1, 0, 1, 1T0, 1T1, 100, 101
  10T  T, 0, 10T, 100
  100  0, 100
  101  0, 1, 100, 101
  11T  T, 0, 1T, 10, 10T, 100, 11T, 110
  110  0, 10, 100, 110

Rémy Sigrist, Table of n, a(n) for n = 0..4688 (rows for n = 0..3^5 flattened)
Wikipedia, Balanced ternary

See A380123 for a similar sequence.

Cf. A005812, A060372, A060373, A368239, A380181.

row(n) = { my (r = [0], d, t = 1); while (n, d = centerlift(Mod(n, 3)); if (d, r = concat(r, [v + d*t | v <- r]);); n = (n-d)/3; t *= 3;); vecsort(r); }

T(n, 1) = - A060373(n).

T(n, 2^A005812(n)) = A060372(n).

A112952 Smaller of two ternary (base 3) numbers (each using only 0's and 1's, the latter's positions never coinciding) such that the decimal representation of their difference is n.

Original entry on oeis.org

0, 0, 1, 0, 0, 11, 10, 10, 1, 0, 0, 1, 0, 0, 111, 110, 110, 101, 100, 100, 101, 100, 100, 11, 10, 10, 1, 0, 0, 1, 0, 0, 11, 10, 10, 1, 0, 0, 1, 0, 0, 1111, 1110, 1110, 1101, 1100, 1100, 1101, 1100, 1100, 1011, 1010, 1010, 1001, 1000, 1000, 1001, 1000, 1000, 1011, 1010

Offset: 0

Lekraj Beedassy, Jan 12 2006

Base 3 representation of A060373. For the larger number see A112867. Any number is expressible as a unique combination of powers of 3 in the form +/- 3^0 +/- 3^1 +/- 3^2 +/- 3^3 ...(Related to Bachet's Weight Problem).

cp's OEIS Frontend

A060372 p(n), positive part of n when n=p(n)-q(n) with p(n), q(n), p(n)+q(n) in A005836, integers written without 2 in base 3.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

A343229 A binary encoding of the digits "-1" in balanced ternary representation of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A060374 a(n)=p+q, where n=p-q and p, q, p+q are in A005836 (integers written without 2 in base 3).

Views

Author

Keywords

Links

Crossrefs

Programs

Haskell

Extensions

A380180 Irregular table T(n, k), n >= 0, k = 1..2^A005812(n); the n-th row lists the integers m (possibly negative) such that the nonzero digits in the balanced ternary expansion of m appear in the balanced ternary expansion of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A112952 Smaller of two ternary (base 3) numbers (each using only 0's and 1's, the latter's positions never coinciding) such that the decimal representation of their difference is n.

Views

Author

Keywords

Comments