A060372 -id:A060372 - cp's OEIS frontend

A060373 q(n), negative 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, 0, 1, 0, 0, 4, 3, 3, 1, 0, 0, 1, 0, 0, 13, 12, 12, 10, 9, 9, 10, 9, 9, 4, 3, 3, 1, 0, 0, 1, 0, 0, 4, 3, 3, 1, 0, 0, 1, 0, 0, 40, 39, 39, 37, 36, 36, 37, 36, 36, 31, 30, 30, 28, 27, 27, 28, 27, 27, 31, 30, 30, 28, 27, 27, 28, 27, 27, 13, 12, 12, 10, 9, 9, 10, 9, 9, 4, 3, 3, 1, 0, 0, 1, 0

Offset: 0

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

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

Cf. A005836, A060372, A060374.

a060373 n = a060372 n - n  -- Reinhard Zumkeller, Jun 09 2012

A343228 A binary encoding of the digits "+1" in balanced ternary representation of n.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 4, 5, 4, 4, 5, 6, 6, 7, 8, 8, 9, 8, 8, 9, 10, 10, 11, 8, 8, 9, 8, 8, 9, 10, 10, 11, 12, 12, 13, 12, 12, 13, 14, 14, 15, 16, 16, 17, 16, 16, 17, 18, 18, 19, 16, 16, 17, 16, 16, 17, 18, 18, 19, 20, 20, 21, 20, 20, 21, 22, 22, 23, 16, 16, 17, 16

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) = A343229(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     1       1          1
   2     2      1T         10
   3     2      10         10
   4     3      11         11
   5     4     1TT        100
   6     4     1T0        100
   7     5     1T1        101
   8     4     10T        100
   9     4     100        100
  10     5     101        101
  11     6     11T        110
  12     6     110        110
  13     7     111        111
  14     8    1TTT       1000
  15     8    1TT0       1000

Rémy Sigrist, Table of n, a(n) for n = 0..6561
Rémy Sigrist, Scatterplot of (a(n), A343229(n)) for n = 0..3^10
Wikipedia, Balanced ternary

Cf. A059095, A060372, A140267, A289813, A289831, A343229, A343230, A343231.

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(A060372(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).

A112867 Greater 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, 1, 10, 10, 11, 100, 100, 101, 100, 100, 101, 110, 110, 111, 1000, 1000, 1001, 1000, 1000, 1001, 1010, 1010, 1011, 1000, 1000, 1001, 1000, 1000, 1001, 1010, 1010, 1011, 1100, 1100, 1101, 1100, 1100, 1101, 1110, 1110, 1111, 10000, 10000, 10001, 10000

Offset: 0

Lekraj Beedassy, Jan 12 2006

Base 3 representation of A060372. For the smaller number see A112952. 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

A060373 q(n), negative 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.

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A343228 A binary encoding of the digits "+1" in balanced ternary representation of n.

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

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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.

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Examples

Links

Crossrefs

Programs

PARI

Formula

A112867 Greater 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.

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Author

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Comments