A344838 -id:A344838 - cp's OEIS frontend

A344834 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = (n * 2^max(0, w(k)-w(n))) AND (k * 2^max(0, w(n)-w(k))) (where AND denotes the bitwise AND operator and w = A070939).

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 4, 2, 2, 4, 0, 0, 4, 4, 3, 4, 4, 0, 0, 4, 4, 4, 4, 4, 4, 0, 0, 4, 4, 4, 4, 4, 4, 4, 0, 0, 8, 4, 6, 4, 4, 6, 4, 8, 0, 0, 8, 8, 6, 4, 5, 4, 6, 8, 8, 0, 0, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 0, 0, 8, 8, 8, 8, 5, 6, 5, 8, 8, 8, 8, 0

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then apply the bitwise AND operator.

			Array T(n, k) begins:
  n\k|  0  1  2   3  4   5   6   7  8  9  10  11  12  13  14  15
  ---+----------------------------------------------------------
    0|  0  0  0   0  0   0   0   0  0  0   0   0   0   0   0   0
    1|  0  1  2   2  4   4   4   4  8  8   8   8   8   8   8   8
    2|  0  2  2   2  4   4   4   4  8  8   8   8   8   8   8   8
    3|  0  2  2   3  4   4   6   6  8  8   8   8  12  12  12  12
    4|  0  4  4   4  4   4   4   4  8  8   8   8   8   8   8   8
    5|  0  4  4   4  4   5   4   5  8  8  10  10   8   8  10  10
    6|  0  4  4   6  4   4   6   6  8  8   8   8  12  12  12  12
    7|  0  4  4   6  4   5   6   7  8  8  10  10  12  12  14  14
    8|  0  8  8   8  8   8   8   8  8  8   8   8   8   8   8   8
    9|  0  8  8   8  8   8   8   8  8  9   8   9   8   9   8   9
   10|  0  8  8   8  8  10   8  10  8  8  10  10   8   8  10  10
   11|  0  8  8   8  8  10   8  10  8  9  10  11   8   9  10  11
   12|  0  8  8  12  8   8  12  12  8  8   8   8  12  12  12  12
   13|  0  8  8  12  8   8  12  12  8  9   8   9  12  13  12  13
   14|  0  8  8  12  8  10  12  14  8  8  10  10  12  12  14  14
   15|  0  8  8  12  8  10  12  14  8  9  10  11  12  13  14  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A004198, A053644, A070939.

Cf. A344835 (OR), A344836 (XOR), A344837 (min), A344838 (max), A344839 (absolute difference).

T(n,k,op=bitand,w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = n.
T(n, 1) = A053644(n).

A344835 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = (n * 2^max(0, w(k)-w(n))) OR (k * 2^max(0, w(n)-w(k))) (where OR denotes the bitwise OR operator and w = A070939).

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 3, 2, 2, 3, 4, 3, 2, 3, 4, 5, 4, 3, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 7, 6, 5, 6, 6, 5, 6, 7, 8, 7, 6, 7, 4, 7, 6, 7, 8, 9, 8, 7, 6, 5, 5, 6, 7, 8, 9, 10, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 10, 9, 12, 7, 7, 7, 7, 12, 9, 10, 11, 12, 11, 10, 13, 8, 7, 6, 7, 8, 13, 10, 11, 12

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then apply the bitwise OR operator.

			Array T(n, k) begins:
  n\k|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
  ---+----------------------------------------------------------------
    0|   0   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    2|   2   2   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    3|   3   3   3   3   6   7   6   7  12  13  14  15  12  13  14  15
    4|   4   4   4   6   4   5   6   7   8   9  10  11  12  13  14  15
    5|   5   5   5   7   5   5   7   7  10  11  10  11  14  15  14  15
    6|   6   6   6   6   6   7   6   7  12  13  14  15  12  13  14  15
    7|   7   7   7   7   7   7   7   7  14  15  14  15  14  15  14  15
    8|   8   8   8  12   8  10  12  14   8   9  10  11  12  13  14  15
    9|   9   9   9  13   9  11  13  15   9   9  11  11  13  13  15  15
   10|  10  10  10  14  10  10  14  14  10  11  10  11  14  15  14  15
   11|  11  11  11  15  11  11  15  15  11  11  11  11  15  15  15  15
   12|  12  12  12  12  12  14  12  14  12  13  14  15  12  13  14  15
   13|  13  13  13  13  13  15  13  15  13  13  15  15  13  13  15  15
   14|  14  14  14  14  14  14  14  14  14  15  14  15  14  15  14  15
   15|  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A003986, A070939.

Cf. A344834 (AND), A344836 (XOR), A344837 (min), A344838 (max), A344839 (absolute difference).

T(n, k, op=bitor, w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = n.
T(n, 1) = max(1, n).

A344836 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = (n * 2^max(0, w(k)-w(n))) XOR (k * 2^max(0, w(n)-w(k))) (where XOR denotes the bitwise XOR operator and w = A070939).

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 0, 0, 3, 4, 1, 0, 1, 4, 5, 0, 1, 1, 0, 5, 6, 1, 0, 0, 0, 1, 6, 7, 2, 1, 2, 2, 1, 2, 7, 8, 3, 2, 3, 0, 3, 2, 3, 8, 9, 0, 3, 0, 1, 1, 0, 3, 0, 9, 10, 1, 0, 1, 2, 0, 2, 1, 0, 1, 10, 11, 2, 1, 4, 3, 3, 3, 3, 4, 1, 2, 11, 12, 3, 2, 5, 0, 2, 0, 2, 0, 5, 2, 3, 12

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then apply the bitwise XOR operator.

			Array T(n, k) begins:
  n\k|   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ---+-------------------------------------------------------
    0|   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
    1|   1  0  0  1  0  1  2  3  0  1   2   3   4   5   6   7
    2|   2  0  0  1  0  1  2  3  0  1   2   3   4   5   6   7
    3|   3  1  1  0  2  3  0  1  4  5   6   7   0   1   2   3
    4|   4  0  0  2  0  1  2  3  0  1   2   3   4   5   6   7
    5|   5  1  1  3  1  0  3  2  2  3   0   1   6   7   4   5
    6|   6  2  2  0  2  3  0  1  4  5   6   7   0   1   2   3
    7|   7  3  3  1  3  2  1  0  6  7   4   5   2   3   0   1
    8|   8  0  0  4  0  2  4  6  0  1   2   3   4   5   6   7
    9|   9  1  1  5  1  3  5  7  1  0   3   2   5   4   7   6
   10|  10  2  2  6  2  0  6  4  2  3   0   1   6   7   4   5
   11|  11  3  3  7  3  1  7  5  3  2   1   0   7   6   5   4
   12|  12  4  4  0  4  6  0  2  4  5   6   7   0   1   2   3
   13|  13  5  5  1  5  7  1  3  5  4   7   6   1   0   3   2
   14|  14  6  6  2  6  4  2  0  6  7   4   5   2   3   0   1
   15|  15  7  7  3  7  5  3  1  7  6   5   4   3   2   1   0

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A003987, A053645, A070939.

Cf. A344834 (AND), A344835 (OR), A344837 (min), A344838 (max), A344839 (absolute difference).

T(n, k, op=bitxor, w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(n, n) = 0.
T(n, 0) = n.
T(n, 1) = A053645(n) for any n > 0.

A344837 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = min(n * 2^max(0, w(k)-w(n)), k * 2^max(0, w(n)-w(k))) (where w = A070939).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 2, 2, 2, 0, 0, 4, 2, 2, 4, 0, 0, 4, 4, 3, 4, 4, 0, 0, 4, 4, 4, 4, 4, 4, 0, 0, 4, 4, 5, 4, 5, 4, 4, 0, 0, 8, 4, 6, 4, 4, 6, 4, 8, 0, 0, 8, 8, 6, 4, 5, 4, 6, 8, 8, 0, 0, 8, 8, 8, 4, 5, 5, 4, 8, 8, 8, 0, 0, 8, 8, 9, 8, 5, 6, 5, 8, 9, 8, 8, 0

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then take the least value.

			Array T(n, k) begins:
  n\k|  0  1  2   3  4   5   6   7  8  9  10  11  12  13  14  15
  ---+----------------------------------------------------------
    0|  0  0  0   0  0   0   0   0  0  0   0   0   0   0   0   0
    1|  0  1  2   2  4   4   4   4  8  8   8   8   8   8   8   8
    2|  0  2  2   2  4   4   4   4  8  8   8   8   8   8   8   8
    3|  0  2  2   3  4   5   6   6  8  9  10  11  12  12  12  12
    4|  0  4  4   4  4   4   4   4  8  8   8   8   8   8   8   8
    5|  0  4  4   5  4   5   5   5  8  9  10  10  10  10  10  10
    6|  0  4  4   6  4   5   6   6  8  9  10  11  12  12  12  12
    7|  0  4  4   6  4   5   6   7  8  9  10  11  12  13  14  14
    8|  0  8  8   8  8   8   8   8  8  8   8   8   8   8   8   8
    9|  0  8  8   9  8   9   9   9  8  9   9   9   9   9   9   9
   10|  0  8  8  10  8  10  10  10  8  9  10  10  10  10  10  10
   11|  0  8  8  11  8  10  11  11  8  9  10  11  11  11  11  11
   12|  0  8  8  12  8  10  12  12  8  9  10  11  12  12  12  12
   13|  0  8  8  12  8  10  12  13  8  9  10  11  12  13  13  13
   14|  0  8  8  12  8  10  12  14  8  9  10  11  12  13  14  14
   15|  0  8  8  12  8  10  12  14  8  9  10  11  12  13  14  15

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A003983, A053644, A070939.

Cf. A344834 (AND), A344835 (OR), A344836 (XOR), A344838 (max), A344839 (absolute difference).

T(n, k, op=min, w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
T(n, 1) = A053644(n).

A344839 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = abs(n * 2^max(0, w(k)-w(n)) - k * 2^max(0, w(n)-w(k))) (where w = A070939).

Original entry on oeis.org

0, 1, 1, 2, 0, 2, 3, 0, 0, 3, 4, 1, 0, 1, 4, 5, 0, 1, 1, 0, 5, 6, 1, 0, 0, 0, 1, 6, 7, 2, 1, 2, 2, 1, 2, 7, 8, 3, 2, 1, 0, 1, 2, 3, 8, 9, 0, 3, 0, 1, 1, 0, 3, 0, 9, 10, 1, 0, 1, 2, 0, 2, 1, 0, 1, 10, 11, 2, 1, 4, 3, 1, 1, 3, 4, 1, 2, 11, 12, 3, 2, 3, 0, 2, 0, 2, 0, 3, 2, 3, 12

Offset: 0

Rémy Sigrist, May 29 2021

In other words, we right pad the binary expansion of the lesser of n and k with zeros (provided it is positive) so that both numbers have the same number of binary digits, and then take the absolute difference.

			Array T(n, k) begins:
  n\k|   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
  ---+-------------------------------------------------------
    0|   0  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15
    1|   1  0  0  1  0  1  2  3  0  1   2   3   4   5   6   7
    2|   2  0  0  1  0  1  2  3  0  1   2   3   4   5   6   7
    3|   3  1  1  0  2  1  0  1  4  3   2   1   0   1   2   3
    4|   4  0  0  2  0  1  2  3  0  1   2   3   4   5   6   7
    5|   5  1  1  1  1  0  1  2  2  1   0   1   2   3   4   5
    6|   6  2  2  0  2  1  0  1  4  3   2   1   0   1   2   3
    7|   7  3  3  1  3  2  1  0  6  5   4   3   2   1   0   1
    8|   8  0  0  4  0  2  4  6  0  1   2   3   4   5   6   7
    9|   9  1  1  3  1  1  3  5  1  0   1   2   3   4   5   6
   10|  10  2  2  2  2  0  2  4  2  1   0   1   2   3   4   5
   11|  11  3  3  1  3  1  1  3  3  2   1   0   1   2   3   4
   12|  12  4  4  0  4  2  0  2  4  3   2   1   0   1   2   3
   13|  13  5  5  1  5  3  1  1  5  4   3   2   1   0   1   2
   14|  14  6  6  2  6  4  2  0  6  5   4   3   2   1   0   1
   15|  15  7  7  3  7  5  3  1  7  6   5   4   3   2   1   0

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10

Cf. A049581, A053645, A070939.

Cf. A344834 (AND), A344835 (OR), A344836 (XOR), A344837 (min), A344838 (max).

T(n,k,op=(x,y)->abs(x-y),w=m->#binary(m)) = { op(n*2^max(0, w(k)-w(n)), k*2^max(0, w(n)-w(k))) }

T(n, k) = T(k, n).
T(n, n) = 0.
T(n, 0) = n.
T(n, 1) = A053645(n) for any n > 0.

cp's OEIS Frontend

A344834 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = (n * 2^max(0, w(k)-w(n))) AND (k * 2^max(0, w(n)-w(k))) (where AND denotes the bitwise AND operator and w = A070939).

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A344835 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = (n * 2^max(0, w(k)-w(n))) OR (k * 2^max(0, w(n)-w(k))) (where OR denotes the bitwise OR operator and w = A070939).

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A344836 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = (n * 2^max(0, w(k)-w(n))) XOR (k * 2^max(0, w(n)-w(k))) (where XOR denotes the bitwise XOR operator and w = A070939).

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A344837 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = min(n * 2^max(0, w(k)-w(n)), k * 2^max(0, w(n)-w(k))) (where w = A070939).

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A344839 Square array T(n, k), n, k >= 0, read by antidiagonals; T(n, k) = abs(n * 2^max(0, w(k)-w(n)) - k * 2^max(0, w(n)-w(k))) (where w = A070939).

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