A343835 -id:A343835 - cp's OEIS frontend

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

Rémy Sigrist, May 01 2021

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.

			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]

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

Cf. A069010, A295989, A343835.

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))) }

T(n, 0) = 0.
T(n, 1) = A342410(n) for any n > 0.
T(n, 2^A069010(n)-1) = n.

A356397 a(n) is the product of the terms in the n-th row of triangle A343835; a(0) = 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 7, 8, 8, 16, 24, 12, 12, 14, 15, 16, 16, 32, 48, 64, 64, 96, 112, 24, 24, 48, 72, 28, 28, 30, 31, 32, 32, 64, 96, 128, 128, 192, 224, 256, 256, 512, 768, 384, 384, 448, 480, 48, 48, 96, 144, 192, 192, 288, 336, 56, 56, 112, 168, 60, 60, 62

Offset: 0

Rémy Sigrist, Aug 05 2022

			For n = 11:
- row 11 of A343835 is (8, 3),
- so a(11) = 8 * 3 = 24.

Rémy Sigrist, Table of n, a(n) for n = 0..8192
Index entries for sequences related to binary expansion of n

Cf. A023758, A059867, A246674, A277561, A343835.

a(n) = { my (v=1); while (n, my (z=valuation(n, 2), o=valuation(n/2^z+1, 2), r=(2^o-1)*2^z); n-=r; v*=r); v }

a(2*n) = a(n) * A277561(n).
a(n) = n iff n belongs to A023758 \ {0}.
A246674(n) divides a(n).

cp's OEIS Frontend

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.

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