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

Author

Rémy Sigrist, May 01 2021

Keywords

• nonn
• base
• tabf
• easy

Comments

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

Examples

			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]
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..6145
• Index entries for sequences related to binary expansion of n

Crossrefs

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

Programs

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 }

Formula

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

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

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