login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A317309 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) has a central valley. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..62.

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

EXAMPLE

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.

PROG

(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

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.

Sequence in context: A129096 A079448 A045407 * A243897 A153075 A287940

Adjacent sequences:  A317306 A317307 A317308 * A317310 A317311 A317312

KEYWORD

nonn

AUTHOR

Omar E. Pol, Aug 29 2018

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 28 06:40 EDT 2021. Contains 346317 sequences. (Running on oeis4.)