A373345 Irregular triangle read by rows where row n lists (in decreasing order) the elements of the Schreier set encoded by A371176(n).

1, 2, 3, 3, 2, 4, 4, 2, 4, 3, 5, 5, 2, 5, 3, 5, 4, 5, 4, 3, 6, 6, 2, 6, 3, 6, 4, 6, 4, 3, 6, 5, 6, 5, 3, 6, 5, 4, 7, 7, 2, 7, 3, 7, 4, 7, 4, 3, 7, 5, 7, 5, 3, 7, 5, 4, 7, 6, 7, 6, 3, 7, 6, 4, 7, 6, 5, 7, 6, 5, 4, 8, 8, 2, 8, 3, 8, 4, 8, 4, 3, 8, 5, 8, 5, 3, 8, 5, 4, 8, 6

Offset: 1

Author

Paolo Xausa, Jun 01 2024

Keywords

• nonn
• tabf
• easy
• base

Comments

A Schreier set is a subset of the positive integers with cardinality less than or equal to the minimum element in the set.

Each term k of A371176 can be put into a one-to-one correspondence with a Schreier set by interpreting the 1-based position of the ones in the binary expansion of k (where position 1 corresponds to the least significant bit) as the elements of the corresponding Schreier set (see A371176 and Bird link).

See A373359 for the elements in each set arranged in increasing order.

The number of sets having maximum element m is A000045(m).

Examples

			Triangle begins:
                                   Corresponding Schreier
   n  A371176(n)  bin(A371176(n))  set (this sequence)
  -------------------------------------------------------
   1      1              1         {1}
   2      2             10         {2}
   3      4            100         {3}
   4      6            110         {3, 2}
   5      8           1000         {4}
   6     10           1010         {4, 2}
   7     12           1100         {4, 3}
   8     16          10000         {5}
   9     18          10010         {5, 2}
  10     20          10100         {5, 3}
  11     24          11000         {5, 4}
  12     28          11100         {5, 4, 3}
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000 (rows 1..2261 of the triangle, flattened).
• Alistair Bird, Jozef Schreier, Schreier sets and the Fibonacci sequence, Out Of The Norm blog, May 13 2012.

Crossrefs

Cf. A000045, A371176, A373359, A373556, A373557.

Cf. A007895 (conjectured row lengths), A072649 (first column), A373346 (row sums), A373347.

Programs

Mathematica

Join[{{1}}, Map[Reverse[PositionIndex[Reverse[IntegerDigits[#, 2]]][1]] &, Select[Range[2, 200, 2], DigitCount[#, 2, 1] <= IntegerExponent[#, 2] + 1 &]]]

Formula

T(n,k) = A373557(n,k) - 1.