A352727 Square array A(n, k), n, k >= 0, read by antidiagonals: the binary expansion of A(n, k) contains the runs of consecutive 1's that appear both in the binary expansions of n and k.

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

Offset

0,13

Comments

We only consider maximal runs of one or more consecutive 1's (as counted by A069010) that completely match in binary expansions of n and k, not simply single common 1's.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k); black pixels denote 0's)

Index entries for sequences related to binary expansion of n

Formula

A(n, k) = A(k, n).

A(n, 0) = 0.

A(n, n) = n.

A(n, 2*n) = 0.

A(n, k) <= A004198(n, k) (bitwise AND operator).

A(n, n+1) = A352729(n).

Example

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

Prog

(PARI) A352724(n) = { my (r=[], o=0); while (n, my (v=valuation(n+n%2, 2)); if (n%2, r=concat(r, (2^v-1)*2^o)); o+=v; n\=2^v); r }
A(n, k) = vecsum(setintersect(A352724(n), A352724(k)))

Crossrefs

Cf. A004198, A069010, A352724, A352729.

Sequence in context: A024155 A251570 A292241 * A332015 A173667 A241541

Adjacent sequences: A352724 A352725 A352726 * A352728 A352729 A352730

Keyword

nonn,base,tabl

Author

Rémy Sigrist, Mar 30 2022

Status

approved