A346872 Irregular triangle read by rows in which row n lists the row 2^(n-1) of A237591, n >= 1.

1, 2, 3, 1, 5, 2, 1, 9, 3, 2, 1, 1, 17, 6, 3, 2, 2, 1, 1, 33, 11, 6, 4, 2, 2, 2, 1, 2, 1, 65, 22, 11, 7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1, 129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 257, 86, 43, 26, 18, 12, 10, 8, 6, 5, 4, 4, 3, 3, 3, 2, 3

Offset: 1

Author

Omar E. Pol, Aug 06 2021

Keywords

• nonn
• tabf

Comments

The characteristic shape of the symmetric representation of sigma(2^(n-1)) consists in that the diagram contains exactly one region (or part) and that region has width 1.

So knowing this characteristic shape we can know if a number is power of 2 or not just by looking at the diagram, even ignoring the concept of power of 2.

Therefore we can see a geometric pattern of the distribution of the powers of 2 in the stepped pyramid described in A245092.

For the definition of "width" see A249351.

T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(2^(n-1)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000079(n-1).

T(n,k) is also the difference between the total number of partitions of all positive integers <= 2^(n-1) into exactly k consecutive parts, and the total number of partitions of all positive integers <= 2^(n-1) into exactly k + 1 consecutive parts.

Examples

			Triangle begins:
    1;
    2;
    3,  1;
    5,  2,  1;
    9,  3,  2,  1, 1;
   17,  6,  3,  2, 2, 1, 1;
   33, 11,  6,  4, 2, 2, 2, 1, 2, 1;
   65, 22, 11,  7, 5, 3, 3, 2, 2, 2, 1, 2, 1, 1, 1;
  129, 43, 22, 13, 9, 7, 5, 4, 3, 3, 3, 2, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1;
...
Illustration of initial terms:
.
Row 1:  _
       |_|                              Semilength = 1
        1
Row 2:    _
        _| |
       |_ _|
         2                              Semilength = 2
.
Row 3:        _
             | |
            _| |
        _ _|  _|
       |_ _ _|1                         Semilength = 4
          3
.
Row 4:                _
                     | |
                     | |
                     | |
                  _ _| |
                _|  _ _|
               |  _|
        _ _ _ _| | 1                    Semilength = 8
       |_ _ _ _ _|2
            5
.
Row 5:                                _
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                     | |
                                _ _ _| |
                               |  _ _ _|
                              _| |
                            _|  _|
                        _ _|  _|        Semilength = 16
                       |  _ _|1 1
                       | | 2
        _ _ _ _ _ _ _ _| |3
       |_ _ _ _ _ _ _ _ _|
                9
.
The area (also the number of cells) of the successive diagrams gives the nonzero Mersenne numbers A000225.
		

Crossrefs

Row sums give A000079.

Column 1 gives A094373.

For the characteristic shape of sigma(A000040(n)) see A346871.

For the characteristic shape of sigma(A000217(n)) see A346873.

For the visualization of Mersenne numbers A000225 see A346874.

For the characteristic shape of sigma(A000384(n)) see A346875.

For the characteristic shape of sigma(A000396(n)) see A346876.

For the characteristic shape of sigma(A008588(n)) see A224613.

Cf. A000203, A000225, A237591, A237593, A245092, A249351, A262626.