

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
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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 zigzag 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



