A317309 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) has a central valley.

3, 5, 11, 13, 23, 37, 41, 43, 59, 61, 79, 83, 89, 107, 109, 113, 137, 139, 149, 151, 173, 179, 181, 211, 223, 227, 229, 257, 263, 269, 271, 307, 311, 313, 317, 353, 359, 367, 373, 409, 419, 421, 431, 433, 467, 479, 487, 491, 541, 547, 557, 599, 601, 607, 613, 617, 619, 673, 677, 683, 691, 701

Offset: 1

Author

Omar E. Pol, Aug 29 2018

Keywords

• nonn

Comments

Except for the first term 3, primes p such that both Dyck paths of the symmetric representation of sigma(p) have a central valley.

Note that the symmetric representation of sigma of an odd prime consists of two perpendicular bars connected by an irregular zig-zag path (see example).

Odd primes and the terms of this sequence are easily identifiable in the pyramid described in A245092 (see Links section).

For more information about the mentioned Dyck paths see A237593.

Equivalently, primes p such that the largest Dyck path of the symmetric representation of sigma(p) has an even number of peaks.

Examples

			Illustration of initial terms:
-------------------------------------------------
   p  sigma(p)  Diagram of the symmetry of sigma
-------------------------------------------------
                     _   _           _   _
                    | | | |         | | | |
                 _ _|_| | |         | | | |
   3      4     |_ _|  _|_|         | | | |
                 _ _ _|             | | | |
   5      6     |_ _ _|             | | | |
                                 _ _|_| | |
                               _|    _ _|_|
                             _|     |
                            |      _|
                 _ _ _ _ _ _|  _ _|
  11     12     |_ _ _ _ _ _| |
                 _ _ _ _ _ _ _|
  13     14     |_ _ _ _ _ _ _|
.
For the first four terms of the sequence we can see in the above diagram that the largest Dyck path of the symmetric representation of sigma(p) has a central valley.
Compare with A317308.
		

Links

• Omar E. Pol, Perspective view of the pyramid (first 16 levels)

Crossrefs

Primes in A161983.

Except for the first term 3, primes in A317304.

The union of A317308 and this sequence gives A000040.

Primes of the triangle of A060300. - César Aguilera, Nov 12 2020

Cf. A000203, A065091, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A239660, A239929, A239931, A239933, A244050, A245092, A262626.

Programs

Python

from sympy import isprime
for x in range(1,100):
     for x in range(2*x**2+2*x-(2*x//2),2*x**2+2*x+(2*x//2)+1):
           if isprime(x):
              print(x, end=', ') # César Aguilera, Nov 12 2020