A341840 -id:A341840 - cp's OEIS frontend

A341839 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the union of R(n) and of R(k).

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

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.

The underlying idea is to break in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.

			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   2   5   5   6   6   9   9  10  10  13  13  14  14
    2|    2   2   2   2   5   5   5   5  10  10  10  10  13  13  13  13
    3|    3   2   2   3   4   5   5   4  11  10  10  11  12  13  13  12
    4|    4   5   5   4   4   5   5   4  11  10  10  11  11  10  10  11
    5|    5   5   5   5   5   5   5   5  10  10  10  10  10  10  10  10
    6|    6   6   5   5   5   5   6   6   9   9  10  10  10  10   9   9
    7|    7   6   5   4   4   5   6   7   8   9  10  11  11  10   9   8
    8|    8   9  10  11  11  10   9   8   8   9  10  11  11  10   9   8
    9|    9   9  10  10  10  10   9   9   9   9  10  10  10  10   9   9
   10|   10  10  10  10  10  10  10  10  10  10  10  10  10  10  10  10
   11|   11  10  10  11  11  10  10  11  11  10  10  11  11  10  10  11
   12|   12  13  13  12  11  10  10  11  11  10  10  11  12  13  13  12
   13|   13  13  13  13  10  10  10  10  10  10  10  10  13  13  13  13
   14|   14  14  13  13  10  10   9   9   9   9  10  10  13  13  14  14
   15|   15  14  13  12  11  10   9   8   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
Index entries for sequences related to binary expansion of n

Cf. A003188, A003987, A005811, A042963, A070939, A101211, A227736, A341840, A341841.

T(n,k) = { my (r=[], v=0); while (n||k, my (w=min(valuation(n+n%2,2), valuation(k+k%2,2))); r=concat(w,r); n\=2^w; k\=2^w); for (k=1, #r, v=(v+k%2)*2^r[k]-k%2); v }

T(n, k) = T(k, n)
T(m, T(n, k)) = T(T(m, n), k).
T(n, n) = n.
T(n, 0) = 0.
A070939(T(n, k)) = max(A070939(n), A070939(k)).
A003188(T(n, k)) = A003188(n) OR A003188(k) (where OR denotes the bitwise OR operator).
T(n, 1) = A042963(ceiling((n+1)/2)).

A341841 Square array T(n, k), n, k >= 0, read by antidiagonals upwards; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) equals R(n) minus R(k).

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 3, 0, 0, 4, 3, 0, 1, 0, 5, 4, 0, 1, 1, 0, 6, 4, 7, 0, 1, 0, 0, 7, 7, 7, 7, 0, 0, 0, 0, 8, 7, 7, 6, 0, 0, 3, 1, 0, 9, 8, 7, 6, 1, 0, 3, 2, 1, 0, 10, 8, 8, 7, 1, 0, 3, 3, 2, 0, 0, 11, 11, 8, 8, 0, 0, 3, 3, 3, 3, 0, 0, 12, 11, 8, 9, 15, 0, 0, 2, 3, 3, 0, 1, 0

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
This sequence uses set subtraction, and is related to:
- A003987 which uses set difference,
- A341839 which uses set union,
- A341840 which uses set intersection.

			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|   1   0   0   1   1   0   0   1  1  0   0   1   1   0   0   1
    2|   2   3   0   1   1   0   3   2  2  3   0   1   1   0   3   2
    3|   3   3   0   0   0   0   3   3  3  3   0   0   0   0   3   3
    4|   4   4   7   7   0   0   3   3  3  3   0   0   7   7   4   4
    5|   5   4   7   6   1   0   3   2  2  3   0   1   6   7   4   5
    6|   6   7   7   6   1   0   0   1  1  0   0   1   6   7   7   6
    7|   7   7   7   7   0   0   0   0  0  0   0   0   7   7   7   7
    8|   8   8   8   8  15  15  15  15  0  0   0   0   7   7   7   7
    9|   9   8   8   9  14  15  15  14  1  0   0   1   6   7   7   6
   10|  10  11   8   9  14  15  12  13  2  3   0   1   6   7   4   5
   11|  11  11   8   8  15  15  12  12  3  3   0   0   7   7   4   4
   12|  12  12  15  15  15  15  12  12  3  3   0   0   0   0   3   3
   13|  13  12  15  14  14  15  12  13  2  3   0   1   1   0   3   2
   14|  14  15  15  14  14  15  15  14  1  0   0   1   1   0   0   1
   15|  15  15  15  15  15  15  15  15  0  0   0   0   0   0   0   0

Rémy Sigrist, Table of n, a(n) for n = 0..10010
Rémy Sigrist, Colored representation of the table for n, k < 2^10
Rémy Sigrist, PARI program for A341841
Index entries for sequences related to binary expansion of n

Cf. A003188, A003987, A070939, A227736, A341839, A341840.

PARI
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See Links section.
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T(n, n) = 0.
T(n, 0) = n.
T(T(n, k), k) = T(n, k).
A070939(T(n, k)) <= A070939(n).
A003188(T(n, k)) = A003188(n) - (A003188(n) AND A003188(k)) (where AND denotes the bitwise AND operator).

A339674 Irregular triangle T(n, k), n, k >= 0, read by rows; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; row n corresponds to the numbers k such that R(k) is included in R(n), in ascending order.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 3, 0, 3, 0, 3, 4, 7, 0, 1, 2, 3, 4, 5, 6, 7, 0, 1, 6, 7, 0, 7, 0, 7, 8, 15, 0, 1, 6, 7, 8, 9, 14, 15, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 0, 3, 4, 7, 8, 11, 12, 15, 0, 3, 12, 15, 0, 1, 2, 3, 12, 13, 14, 15, 0, 1, 14, 15, 0

Offset: 0

Rémy Sigrist, Feb 21 2021

For any m > 0, R(m) contains the partial sums of the m-th row of A227736; by convention, R(0) = {}.
The underlying idea is to take some or all of the rightmost runs of a number, and possibly merge some of them.
For any n >= 0, the n-th row:
- has 2^A000120(A003188(n)) terms,
- has first term 0 and last term A003817(n),
- has n at position A090079(n),
- corresponds to the distinct terms in n-th row of table A341840.

			The triangle starts:
    0;
    0, 1;
    0, 1, 2, 3;
    0, 3;
    0, 3, 4, 7;
    0, 1, 2, 3, 4, 5, 6, 7;
    0, 1, 6, 7;
    0, 7;
    0, 7, 8, 15;
    0, 1, 6, 7, 8, 9, 14, 15;
    0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15;
    0, 3, 4, 7, 8, 11, 12, 15;
    0, 3, 12, 15;
    0, 1, 2, 3, 12, 13, 14, 15;
    0, 1, 14, 15;
    0, 15;
    ...

Rémy Sigrist, Table of n, a(n) for n = 0..6560
Rémy Sigrist, Scatterplot of (n, T(n, k)) for n <= 2^10
Rémy Sigrist, PARI program for A339674
Index entries for sequences related to binary expansion of n

Cf. A000120, A003188, A003817, A090079, A227736, A341840.

PARI
```
See Links section.
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T(n, 0) = 0.
T(n, A090079(n)) = n.
T(n, 2^A000120(A003188(n))-1) = A003817(n).

cp's OEIS Frontend

A341839 Square array T(n, k), n, k >= 0, read by antidiagonals; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) is the union of R(n) and of R(k).

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A341841 Square array T(n, k), n, k >= 0, read by antidiagonals upwards; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; T(n, k) is the unique number t such that R(t) equals R(n) minus R(k).

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A339674 Irregular triangle T(n, k), n, k >= 0, read by rows; for any number m with runs in binary expansion (r_1, ..., r_j), let R(m) = {r_1 + ... + r_j, r_2 + ... + r_j, ..., r_j}; row n corresponds to the numbers k such that R(k) is included in R(n), in ascending order.

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