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