A004488 -id:A004488 - cp's OEIS frontend

A253586 The sum of the i-th ternary digits of n, k, and A(n,k) equals 0 (mod 3) for each i>=0 (leading zeros included); square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 2, 2, 1, 1, 1, 6, 0, 0, 6, 8, 8, 2, 8, 8, 7, 7, 7, 7, 7, 7, 3, 6, 6, 3, 6, 6, 3, 5, 5, 8, 5, 5, 8, 5, 5, 4, 4, 4, 4, 4, 4, 4, 4, 4, 18, 3, 3, 0, 3, 3, 0, 3, 3, 18, 20, 20, 5, 2, 2, 5, 2, 2, 5, 20, 20, 19, 19, 19, 1, 1, 1, 1, 1, 1, 19, 19, 19, 24, 18, 18, 24, 0, 0, 6, 0, 0, 24, 18, 18, 24

Offset: 0

Alois P. Heinz, Jan 04 2015

			Square array A(n,k) begins:
  0, 2, 1, 6, 8, 7, 3, 5, 4, ...
  2, 1, 0, 8, 7, 6, 5, 4, 3, ...
  1, 0, 2, 7, 6, 8, 4, 3, 5, ...
  6, 8, 7, 3, 5, 4, 0, 2, 1, ...
  8, 7, 6, 5, 4, 3, 2, 1, 0, ...
  7, 6, 8, 4, 3, 5, 1, 0, 2, ...
  3, 5, 4, 0, 2, 1, 6, 8, 7, ...
  5, 4, 3, 2, 1, 0, 8, 7, 6, ...
  4, 3, 5, 1, 0, 2, 7, 6, 8, ...

Alois P. Heinz, Antidiagonals n = 0..200, flattened
Rémy Sigrist, Colored representation of the table for 0 <= n, k < 3^7 (where the hue is function of T(n, k))
Wikipedia, Set (game)

Column k=0 and row n=0 gives A004488.
Main diagonal gives A001477.
A(n,floor(n/3)) gives A060587.
Cf. A090245, A253587.

A:= proc(n, k) local i, j; `if`(n=0 and k=0, 0,
      A(iquo(n, 3, 'i'), iquo(k, 3, 'j'))*3 +irem(6-i-j, 3))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);

A(n,k) = A(floor(n/3),floor(k/3))*3+(6-(n mod 3)-(k mod 3) mod 3), A(0,0) = 0.

A253587 The sum of the i-th ternary digits of n, k, and T(n,k) equals 0 (mod 3) for each i>=0 (leading zeros included); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 2, 1, 1, 0, 2, 6, 8, 7, 3, 8, 7, 6, 5, 4, 7, 6, 8, 4, 3, 5, 3, 5, 4, 0, 2, 1, 6, 5, 4, 3, 2, 1, 0, 8, 7, 4, 3, 5, 1, 0, 2, 7, 6, 8, 18, 20, 19, 24, 26, 25, 21, 23, 22, 9, 20, 19, 18, 26, 25, 24, 23, 22, 21, 11, 10, 19, 18, 20, 25, 24, 26, 22, 21, 23, 10, 9, 11

Offset: 0

Alois P. Heinz, Jan 04 2015

			Triangle T(n,k) begins:
  0;
  2, 1;
  1, 0, 2;
  6, 8, 7, 3;
  8, 7, 6, 5, 4;
  7, 6, 8, 4, 3, 5;
  3, 5, 4, 0, 2, 1, 6;
  5, 4, 3, 2, 1, 0, 8, 7;
  4, 3, 5, 1, 0, 2, 7, 6, 8;

Alois P. Heinz, Rows n = 0..200, flattened
Wikipedia, Set (game)

Column k=0 gives A004488.
Main diagonal gives A001477.
T(n,floor(n/3)) gives A060587.
Cf. A090245, A253586.

T:= proc(n, k) local i, j; `if`(n=0 and k=0, 0,
      T(iquo(n, 3, 'i'), iquo(k, 3, 'j'))*3 +irem(6-i-j, 3))
    end:
seq(seq(T(n, k), k=0..n), n=0..14);

T(n,k) = T(floor(n/3),floor(k/3))*3+(6-(n mod 3)-(k mod 3) mod 3), T(0,0) = 0.

A361818 For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; this sequence lists the numbers k such that T_k has 3-fold rotational symmetry.

Original entry on oeis.org

0, 1, 2, 4, 8, 13, 26, 34, 40, 46, 59, 65, 80, 112, 121, 130, 224, 233, 242, 304, 364, 424, 518, 578, 728, 772, 862, 925, 1003, 1093, 1183, 1261, 1324, 1414, 1535, 1598, 1688, 1766, 1856, 1919, 2006, 2096, 2186, 2257, 2509, 2734, 3028, 3280, 3532, 3826, 4051

Offset: 1

Rémy Sigrist, Mar 25 2023

We can devise a similar sequence for any fixed base b >= 2; the present sequence corresponds to b = 3, and A334556 corresponds to b = 2.
This sequence is infinite as it contains A048328.
If k belongs to the sequence, then A004488(k) and A030102(k) belong to the sequence.
Empirically, there are 2*3^floor((w-1)/3) positive terms with w ternary digits.
For any k, if t appears above u and v in T_k, then t + u + v = 0 (mod 3) and #{t, u, v} = 1 or 3 (the three values are either equal or all distinct); each value is uniquely determined by the two others in the same way: t = (-u-v) mod 3, u = (-t-v) mod 3, v = (-t-u) mod 3; this means that we can reconstruct T_k from any of its three sides.
If some row of T_k, say r, has w values and corresponds to the ternary expansion of m, then the row above r corresponds to the w-1 rightmost digits of the ternary expansion of A060587(m).
All positive terms belong to A297250 (their most significant digit equals their least significant digit in base 3).

			The ternary expansion of 304 is "102021", and the corresponding triangle is:
             1
            0 2
           2 1 0
          0 1 1 2
         2 1 1 1 0
        1 0 2 0 2 1
As this triangle has 3-fold rotational symmetry, 304 belongs to the sequence.

Rémy Sigrist, Triangles illustrating initial terms
Rémy Sigrist, PARI program
Index entries for sequences related to XOR-triangles

Cf. A004488, A048328, A060587, A297250, A334556.

PARI
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See Links section.
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A380349 In the ternary expansion of n, from left to right: replace the first, third, fifth, etc. nonzero digit, says d, by 3-d.

Original entry on oeis.org

0, 2, 1, 6, 7, 8, 3, 4, 5, 18, 19, 20, 21, 23, 22, 24, 26, 25, 9, 10, 11, 12, 14, 13, 15, 17, 16, 54, 55, 56, 57, 59, 58, 60, 62, 61, 63, 65, 64, 69, 70, 71, 66, 67, 68, 72, 74, 73, 78, 79, 80, 75, 76, 77, 27, 28, 29, 30, 32, 31, 33, 35, 34, 36, 38, 37, 42, 43

Offset: 0

Rémy Sigrist, Jan 22 2025

This sequence is a self-inverse permutation of the nonnegative integers.

			The ternary expansion of 321 is "102220", so the ternary expansion of a(321) is "202120", and a(321) = 555.

See A380350, A380351 and A380352 for similar sequences.

Cf. A004488.

a(n) = { my (d = digits(n, 3), nz = 0); for (k = 1, #d, if (d[k], if (nz++%2==1, d[k] = 3-d[k];););); fromdigits(d, 3); }

from gmpy2 import digits
def a(n):
    d, nz = list(digits(n, 3)), 0
    for i, di in enumerate(d):
        if di != '0':
            nz += 1
            if nz&1: d[i] = '2' if di == '1' else '1'
    return int("".join(d), 3)
print([a(n) for n in range(68)]) # Michael S. Branicky, Jan 24 2025

a(A380350(n)) = A380350(a(n)) = A004488.

{a(n), A380350(n)} = {A380351(n), A380352(n)}.

A380350 In the ternary expansion of n, from left to right: replace the second, fourth, sixth, etc. nonzero digit, says d, by 3-d.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 8, 7, 9, 11, 10, 15, 16, 17, 12, 13, 14, 18, 20, 19, 24, 25, 26, 21, 22, 23, 27, 29, 28, 33, 34, 35, 30, 31, 32, 45, 46, 47, 48, 50, 49, 51, 53, 52, 36, 37, 38, 39, 41, 40, 42, 44, 43, 54, 56, 55, 60, 61, 62, 57, 58, 59, 72, 73, 74, 75, 77

Offset: 0

Rémy Sigrist, Jan 22 2025

This sequence is a self-inverse permutation of the nonnegative integers.

			The ternary expansion of 321 is "102220", so the ternary expansion of a(321) is "101210", and a(321) = 292.

See A380349, A380351 and A380352 for similar sequences.

Cf. A004488, A038754.

a(n) = { my (d = digits(n, 3), nz = 0); for (k = 1, #d, if (d[k], if (nz++%2==0, d[k] = 3-d[k];););); fromdigits(d, 3); }

from gmpy2 import digits
def a(n):
    d, nz = list(digits(n, 3)), 0
    for i, di in enumerate(d):
        if di != '0':
            nz += 1
            if nz&1 == 0: d[i] = '2' if di == '1' else '1'
    return int("".join(d), 3)
print([a(n) for n in range(68)]) # Michael S. Branicky, Jan 24 2025

a(A380349(n)) = A380349(a(n)) = A004488.
{a(n), A380349(n)} = {A380351(n), A380352(n)}.
a(n) = n iff n = 0 or n belongs to A038754.

A380351 In the ternary expansion of n, from right to left: replace the first, third, fifth, etc. nonzero digit, says d, by 3-d.

Original entry on oeis.org

0, 2, 1, 6, 5, 4, 3, 8, 7, 18, 11, 10, 15, 23, 22, 12, 26, 25, 9, 20, 19, 24, 14, 13, 21, 17, 16, 54, 29, 28, 33, 59, 58, 30, 62, 61, 45, 65, 64, 69, 50, 49, 66, 53, 52, 36, 74, 73, 78, 41, 40, 75, 44, 43, 27, 56, 55, 60, 32, 31, 57, 35, 34, 72, 38, 37, 42, 77

Offset: 0

Rémy Sigrist, Jan 22 2025

This sequence is a self-inverse permutation of the nonnegative integers.

			The ternary expansion of 321 is "102220", so the ternary expansion of a(321) is "101210", and a(321) = 291.

See A380349, A380350 and A380352 for similar sequences.

Cf. A004488.

a(n) = { my (d = digits(n, 3), nz = 0); forstep (k = #d, 1, -1, if (d[k], if (nz++%2==1, d[k] = 3-d[k];););); fromdigits(d, 3); }

from gmpy2 import digits
def a(n):
    d, nz = list(digits(n, 3)), 0
    for i, di in enumerate(d[::-1], 1):
        if di != '0':
            nz += 1
            if nz&1: d[-i] = '2' if di == '1' else '1'
    return int("".join(d), 3)
print([a(n) for n in range(68)]) # Michael S. Branicky, Jan 24 2025

a(A380352(n)) = A380352(a(n)) = A004488(n).

{a(n), A380352(n)} = {A380349(n), A380350(n)}.

A380352 In the ternary expansion of n, from right to left: replace the second, fourth, sixth, etc. nonzero digit, says d, by 3-d.

Original entry on oeis.org

0, 1, 2, 3, 7, 8, 6, 4, 5, 9, 19, 20, 21, 16, 17, 24, 13, 14, 18, 10, 11, 12, 25, 26, 15, 22, 23, 27, 55, 56, 57, 34, 35, 60, 31, 32, 63, 46, 47, 48, 70, 71, 51, 67, 68, 72, 37, 38, 39, 79, 80, 42, 76, 77, 54, 28, 29, 30, 61, 62, 33, 58, 59, 36, 73, 74, 75, 43

Offset: 0

Rémy Sigrist, Jan 22 2025

This sequence is a self-inverse permutation of the nonnegative integers.

			The ternary expansion of 321 is "102220", so the ternary expansion of a(321) is "202120", and a(321) = 555.

See A380349, A380350 and A380351 for similar sequences.

Cf. A004488, A038754.

a(n) = { my (d = digits(n, 3), nz = 0); forstep (k = #d, 1, -1, if (d[k], if (nz++%2==0, d[k] = 3-d[k];););); fromdigits(d, 3); }

from gmpy2 import digits
def a(n):
    d, nz = list(digits(n, 3)), 0
    for i, di in enumerate(d[::-1], 1):
        if di != '0':
            nz += 1
            if nz&1 == 0: d[-i] = '2' if di == '1' else '1'
    return int("".join(d), 3)
print([a(n) for n in range(68)]) # Michael S. Branicky, Jan 24 2025

a(A380351(n)) = A380351(a(n)) = A004488(n).
{a(n), A380351(n)} = {A380349(n), A380350(n)}.
a(n) = n iff n = 0 or n belongs to A038754.

A055116 Base-6 complement of n (write n in base 6, then replace each digit with its base-6 negative).

Original entry on oeis.org

0, 5, 4, 3, 2, 1, 30, 35, 34, 33, 32, 31, 24, 29, 28, 27, 26, 25, 18, 23, 22, 21, 20, 19, 12, 17, 16, 15, 14, 13, 6, 11, 10, 9, 8, 7, 180, 185, 184, 183, 182, 181, 210, 215, 214, 213, 212, 211, 204, 209, 208, 207, 206, 205, 198, 203, 202, 201, 200, 199, 192, 197, 196

Offset: 0

Henry Bottomley, Apr 19 2000

Cf. A004488, A048647, A055115-A055126.

Column k=6 of A248813.

A289838 a(n) = A289815(n) * A289816(n).

Original entry on oeis.org

1, 2, 2, 3, 6, 6, 3, 6, 6, 4, 10, 10, 12, 30, 30, 12, 30, 30, 4, 10, 10, 12, 30, 30, 12, 30, 30, 5, 14, 14, 15, 42, 42, 15, 42, 42, 20, 70, 70, 60, 210, 210, 60, 210, 210, 20, 70, 70, 60, 210, 210, 60, 210, 210, 5, 14, 14, 15, 42, 42, 15, 42, 42, 20, 70, 70

Offset: 0

Rémy Sigrist, Jul 13 2017

Each number k > 0 appears 2^omega(k) times (where omega = A001221).
a(A004488(n)) = a(n) for any n >= 0.
The number of distinct prime factors of a(n) equals the number of nonzero digits in the ternary representation of n.

			a(42) = A289815(42) * A289816(42) = 20 * 3 = 60.

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Cf. A001221, A004488, A289815, A289816.

a(n) = { my (v=1);
       for (o=2, oo,
           if (n==0, return (v));
           if (gcd(v, o)==1 && omega(o)==1,
               if (n % 3, v *= o);
               n \= 3;
           );
       );}

from sympy import gcd, primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    v, o = 1, 2
    while True:
        if n==0: return v
        if gcd(v, o)==1 and omega(o)==1:
            if n%3: v*=o
            n //= 3
        o+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 02 2017

A325827 a(n) = A325825(2*n, sigma(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 4, 1, 4, 3, 1, 1, 3, 1, 1, 1, 4, 1, 3, 5, 1, 1, 28, 1, 3, 1, 1, 3, 1, 1, 4, 1, 1, 1, 10, 14, 12, 1, 4, 3, 1, 1, 4, 5, 1, 3, 5, 1, 12, 1, 4, 1, 1, 1, 12, 5, 4, 1, 1, 1, 48, 1, 1, 3, 16, 1, 3, 5, 1, 5, 4, 1, 12, 1, 1, 1, 14, 1, 28, 1, 1, 3, 1, 1, 9, 16, 1, 5, 1, 1, 12, 5, 11, 3, 1, 34, 12, 1, 5, 3

Offset: 1

Antti Karttunen, May 22 2019

nonn

cp's OEIS Frontend

A253586 The sum of the i-th ternary digits of n, k, and A(n,k) equals 0 (mod 3) for each i>=0 (leading zeros included); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A253587 The sum of the i-th ternary digits of n, k, and T(n,k) equals 0 (mod 3) for each i>=0 (leading zeros included); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A361818 For any number k >= 0, let T_k be the triangle whose base corresponds to the ternary expansion of k (without leading zeros) and other values, say t above u and v, satisfy t = (-u-v) mod 3; this sequence lists the numbers k such that T_k has 3-fold rotational symmetry.

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PARI

A380349 In the ternary expansion of n, from left to right: replace the first, third, fifth, etc. nonzero digit, says d, by 3-d.

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A380350 In the ternary expansion of n, from left to right: replace the second, fourth, sixth, etc. nonzero digit, says d, by 3-d.

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PARI

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A380351 In the ternary expansion of n, from right to left: replace the first, third, fifth, etc. nonzero digit, says d, by 3-d.

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PARI

Python

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A380352 In the ternary expansion of n, from right to left: replace the second, fourth, sixth, etc. nonzero digit, says d, by 3-d.

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A055116 Base-6 complement of n (write n in base 6, then replace each digit with its base-6 negative).