A317309 -id:A317309 - cp's OEIS frontend

A317308 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) has a central peak.

2, 7, 17, 19, 29, 31, 47, 53, 67, 71, 73, 97, 101, 103, 127, 131, 157, 163, 167, 191, 193, 197, 199, 233, 239, 241, 251, 277, 281, 283, 293, 331, 337, 347, 349, 379, 383, 389, 397, 401, 439, 443, 449, 457, 461, 463, 499, 503, 509, 521, 523, 563, 569, 571, 577, 587, 593, 631, 641, 643, 647, 653, 659, 661

Offset: 1

Omar E. Pol, Aug 29 2018

nonn

Also primes p such that both Dyck paths of the symmetric representation of sigma(p) have a central peak.
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 odd number of peaks.

			Illustration of initial terms:
--------------------------------------------------------
   p   sigma(p)   Diagram of the symmetry of sigma
--------------------------------------------------------
                    _         _                   _   _
                  _| |       | |                 | | | |
   2      3      |_ _|       | |                 | | | |
                             | |                 | | | |
                            _|_|                 | | | |
                          _|                     | | | |
                  _ _ _ _|                       | | | |
   7      8      |_ _ _ _|                       | | | |
                                                 | | | |
                                            _ _ _|_| | |
                                           |    _ _ _|_|
                                          _|   |
                                        _|  _ _|
                                    _ _|  _|
                                   |     |
                                   |  _ _|
                  _ _ _ _ _ _ _ _ _| |
  17     18      |_ _ _ _ _ _ _ _ _| |
                  _ _ _ _ _ _ _ _ _ _|
  19     20      |_ _ _ _ _ _ _ _ _ _|
.
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 peak.
Compare with A317309.

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

Primes in A162917.
Also primes in A317303.
The union of this sequence and A317309 gives A000040.
Cf. A000203, A065091, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A239660, A239929, A239931, A239933, A244050, A245092, A262626.

cp's OEIS Frontend

A317308 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) has a central peak.

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