A346871 Irregular triangle read by rows in which row n lists the row A000040(n) of A237591, n >= 1.

2, 2, 1, 3, 2, 4, 2, 1, 6, 3, 1, 1, 7, 3, 2, 1, 9, 4, 2, 1, 1, 10, 4, 2, 2, 1, 12, 5, 2, 2, 1, 1, 15, 6, 3, 2, 1, 1, 1, 16, 6, 3, 2, 2, 1, 1, 19, 7, 4, 2, 2, 1, 1, 1, 21, 8, 4, 2, 2, 2, 1, 1, 22, 8, 4, 3, 2, 1, 2, 1, 24, 9, 4, 3, 2, 2, 1, 1, 1, 27, 10, 5, 3, 2, 2, 1, 2, 1

Offset: 1

Author

Omar E. Pol, Aug 06 2021

Keywords

• nonn
• tabf

Comments

The characteristic shape of the symmetric representation of sigma(prime(n)) consists in that the diagram contains exactly two regions (or parts) and each region is a rectangle (or bar), except for the first prime number (the 2) whose symmetric representation of sigma(2) consists of only one region which contains three cells.

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

Therefore we can see a geometric pattern of the exact distribution of prime numbers in the stepped pyramid described in A245092.

T(n,k) is the length of the k-th line segment of the largest Dyck path of the symmetric representation of sigma(prime(n)), from the border to the center, hence the sum of the n-th row of triangle is equal to A000040(n).

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

Examples

			Triangle begins:
   2;
   2, 1;
   3, 2;
   4, 2, 1;
   6, 3, 1, 1;
   7, 3, 2, 1;
   9, 4, 2, 1, 1;
  10, 4, 2, 2, 1;
  12, 5, 2, 2, 1, 1;
  15, 6, 3, 2, 1, 1, 1;
  16, 6, 3, 2, 2, 1, 1;
  19, 7, 4, 2, 2, 1, 1, 1;
  21, 8, 4, 2, 2, 2, 1, 1;
  22, 8, 4, 3, 2, 1, 2, 1;
  24, 9, 4, 3, 2, 2, 1, 1, 1;
...
Illustration of initial terms:
Row 1:    _
        _| |
       |_ _|
         2                         Semilength = 2
.
Row 2:      _
           | |
        _ _|_|
       |_ _|1                      Semilength = 3
         2
.
Row 3:          _
               | |
               | |
              _|_|
        _ _ _|                     Semilength = 5
       |_ _ _|2
          3
.
Row 4:              _
                   | |
                   | |
                   | |
                  _|_|
                _|
        _ _ _ _| 1                 Semilength = 7
       |_ _ _ _|2
           4
.
Row 5:                         _
                              | |
                              | |
                              | |
                              | |
                              | |
                           _ _|_|
                         _|
                       _|1         Semilength = 11
                      |1
           _ _ _ _ _ _|
          |_ _ _ _ _ _|3
                6
.
The area (also the number of cells) of the successive diagrams gives A008864.
		

Crossrefs

Row sums give A000040.

For the characteristic shape of sigma(A000079(n)) see A346872.

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, A008864, A018252, A237591, A237593, A245092, A249351, A262626.