A343757 Irregular table read by rows; the n-th row contains the sums of distinct terms of the n-th row of table A343835, in ascending order.

0, 0, 1, 0, 2, 0, 3, 0, 4, 0, 1, 4, 5, 0, 6, 0, 7, 0, 8, 0, 1, 8, 9, 0, 2, 8, 10, 0, 3, 8, 11, 0, 12, 0, 1, 12, 13, 0, 14, 0, 15, 0, 16, 0, 1, 16, 17, 0, 2, 16, 18, 0, 3, 16, 19, 0, 4, 16, 20, 0, 1, 4, 5, 16, 17, 20, 21, 0, 6, 16, 22, 0, 7, 16, 23, 0, 24

Offset: 0

Author

Rémy Sigrist, May 01 2021

Keywords

• nonn
• base
• tabf

Comments

In other words, the n-th row contains the numbers k whose runs of 1's in the binary expansion also appear in that of n.

The n-th row has 2^A069010(n) terms.

This sequence has similarities with A295989.

Examples

			Table begins:
     0:    [0]
     1:    [0, 1]
     2:    [0, 2]
     3:    [0, 3]
     4:    [0, 4]
     5:    [0, 1, 4, 5]
     6:    [0, 6]
     7:    [0, 7]
     8:    [0, 8]
     9:    [0, 1, 8, 9]
    10:    [0, 2, 8, 10]
    11:    [0, 3, 8, 11]
    12:    [0, 12]
    13:    [0, 1, 12, 13]
    14:    [0, 14]
    15:    [0, 15]
Table begins in binary:
       0:   [0]
       1:   [0, 1]
      10:   [0, 10]
      11:   [0, 11]
     100:   [0, 100]
     101:   [0, 1, 100, 101]
     110:   [0, 110]
     111:   [0, 111]
    1000:   [0, 1000]
    1001:   [0, 1, 1000, 1001]
    1010:   [0, 10, 1000, 1010]
    1011:   [0, 11, 1000, 1011]
    1100:   [0, 1100]
    1101:   [0, 1, 1100, 1101]
    1110:   [0, 1110]
    1111:   [0, 1111]
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8120
• Rémy Sigrist, Scatterplot of (n, T(n, k)) for n < 2^10
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A069010, A295989, A343835.

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(rr, r)); vector(2^#rr, k, vecsum(vecextract(rr, k-1))) }

Formula

T(n, 0) = 0.

T(n, 1) = A342410(n) for any n > 0.

T(n, 2^A069010(n)-1) = n.