cp's OEIS Frontend

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

%I A317309 #43 Nov 26 2020 23:36:27
%S A317309 3,5,11,13,23,37,41,43,59,61,79,83,89,107,109,113,137,139,149,151,173,
%T A317309 179,181,211,223,227,229,257,263,269,271,307,311,313,317,353,359,367,
%U A317309 373,409,419,421,431,433,467,479,487,491,541,547,557,599,601,607,613,617,619,673,677,683,691,701
%N A317309 Primes p such that the largest Dyck path of the symmetric representation of sigma(p) has a central valley.
%C A317309 Except for the first term 3, primes p such that both Dyck paths of the symmetric representation of sigma(p) have a central valley.
%C A317309 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).
%C A317309 Odd primes and the terms of this sequence are easily identifiable in the pyramid described in A245092 (see Links section).
%C A317309 For more information about the mentioned Dyck paths see A237593.
%C A317309 Equivalently, primes p such that the largest Dyck path of the symmetric representation of sigma(p) has an even number of peaks.
%H A317309 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpyr05.jpg">Perspective view of the pyramid (first 16 levels)</a>
%e A317309 Illustration of initial terms:
%e A317309 -------------------------------------------------
%e A317309    p  sigma(p)  Diagram of the symmetry of sigma
%e A317309 -------------------------------------------------
%e A317309                      _   _           _   _
%e A317309                     | | | |         | | | |
%e A317309                  _ _|_| | |         | | | |
%e A317309    3      4     |_ _|  _|_|         | | | |
%e A317309                  _ _ _|             | | | |
%e A317309    5      6     |_ _ _|             | | | |
%e A317309                                  _ _|_| | |
%e A317309                                _|    _ _|_|
%e A317309                              _|     |
%e A317309                             |      _|
%e A317309                  _ _ _ _ _ _|  _ _|
%e A317309   11     12     |_ _ _ _ _ _| |
%e A317309                  _ _ _ _ _ _ _|
%e A317309   13     14     |_ _ _ _ _ _ _|
%e A317309 .
%e A317309 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.
%e A317309 Compare with A317308.
%o A317309 (Python)
%o A317309 from sympy import isprime
%o A317309 for x in range(1,100):
%o A317309      for x in range(2*x**2+2*x-(2*x//2),2*x**2+2*x+(2*x//2)+1):
%o A317309            if isprime(x):
%o A317309               print(x, end=', ') # _César Aguilera_, Nov 12 2020
%Y A317309 Primes in A161983.
%Y A317309 Except for the first term 3, primes in A317304.
%Y A317309 The union of A317308 and this sequence gives A000040.
%Y A317309 Primes of the triangle of A060300. - _César Aguilera_, Nov 12 2020
%Y A317309 Cf. A000203, A065091, A196020, A236104, A235791, A237048, A237591, A237593, A237270, A239660, A239929, A239931, A239933, A244050, A245092, A262626.
%K A317309 nonn
%O A317309 1,1
%A A317309 _Omar E. Pol_, Aug 29 2018