A341840 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 intersection of R(n) and of R(k).

0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 0, 0, 0, 0, 3, 3, 0, 0, 0, 1, 3, 3, 3, 1, 0, 0, 1, 2, 3, 3, 2, 1, 0, 0, 0, 1, 3, 4, 3, 1, 0, 0, 0, 0, 0, 0, 4, 4, 0, 0, 0, 0, 0, 1, 0, 0, 7, 5, 7, 0, 0, 1, 0, 0, 1, 1, 0, 7, 6, 6, 7, 0, 1, 1, 0, 0, 0, 2, 0, 7, 7, 6, 7, 7, 0, 2, 0, 0

Offset

0,13

Comments

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 merge in an optimal way the runs in binary expansions of n and of k so that they match, hence the relationship with A003188.

Links

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, Colored representation of the table for n, k < 2^10 (black pixels correspond to 0's)

Rémy Sigrist, PARI program for A341840

Index entries for sequences related to binary expansion of n

Formula

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)) <= min(A070939(n), A070939(k)).

A003188(T(n, k)) = A003188(n) AND A003188(k) (where AND denotes the bitwise AND operator).

Example

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

Prog

(PARI) See Links section.

Crossrefs

Cf. A003188, A003987, A005811, A070939, A227736, A341839, A341840, A341841.

Sequence in context: A319072 A348271 A083804 * A028597 A353753 A028617

Adjacent sequences: A341837 A341838 A341839 * A341841 A341842 A341843

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Feb 21 2021

Status

approved