A343835 Irregular table T(n, k), n > 0, k = 1..A069010(n), read by rows; the n-th row contains the shortest partition of n whose values belong to A023758 and can be added without carriers in binary, in descending order.

1, 2, 3, 4, 4, 1, 6, 7, 8, 8, 1, 8, 2, 8, 3, 12, 12, 1, 14, 15, 16, 16, 1, 16, 2, 16, 3, 16, 4, 16, 4, 1, 16, 6, 16, 7, 24, 24, 1, 24, 2, 24, 3, 28, 28, 1, 30, 31, 32, 32, 1, 32, 2, 32, 3, 32, 4, 32, 4, 1, 32, 6, 32, 7, 32, 8, 32, 8, 1, 32, 8, 2, 32, 8, 3

Offset

1,2

Comments

In other words, the n-th row gives the numerical values of the runs of 1's in the binary expansion of n.

Links

Rémy Sigrist, Table of n, a(n) for n = 1..6145

Index entries for sequences related to binary expansion of n

Formula

T(n, 1) = A342126(n).

T(n, A069010(n)) = A342410(n).

Sum_{k = 1..A069010(n)} T(n, k) = n.

Example

Table begins:
     1:   [1]
     2:   [2]
     3:   [3]
     4:   [4]
     5:   [4, 1]
     6:   [6]
     7:   [7]
     8:   [8]
     9:   [8, 1]
    10:   [8, 2]
    11:   [8, 3]
    12:   [12]
    13:   [12, 1]
    14:   [14]
    15:   [15]
Table begins in binary:
       1:   [1]
      10:   [10]
      11:   [11]
     100:   [100]
     101:   [100, 1]
     110:   [110]
     111:   [111]
    1000:   [1000]
    1001:   [1000, 1]
    1010:   [1000, 10]
    1011:   [1000, 11]
    1100:   [1100]
    1101:   [1100, 1]
    1110:   [1110]
    1111:   [1111]

Prog

(PARI) row(n) = { my (rr=[]); while (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2), r=(2^o-1)*2^z); n-=r; rr = concat(r, rr); ); rr }

Crossrefs

Cf. A023758, A069010, A272011, A342126, A342410, A343757.

Sequence in context: A346617 A160386 A347860 * A307310 A339072 A174015

Adjacent sequences: A343832 A343833 A343834 * A343836 A343837 A343838

Keyword

nonn,base,tabf,easy

Author

Rémy Sigrist, May 01 2021

Status

approved