A356477 a(n) is the start of the first sequence of 2*n+1 consecutive primes p_1, p_2, ..., p_(2*n+1) such that p_1*p_2 + p_2*p_3 + ... + p_(2*n)*p_(2*n+1) + p_(2*n+1)*p_1 is prime.

2, 19, 19, 2, 23, 2, 7, 7, 2, 5, 113, 5, 29, 13, 67, 53, 11, 11, 5, 23, 7, 43, 5, 2, 31, 73, 13, 3, 89, 5, 11, 3, 89, 31, 43, 2, 37, 2, 23, 7, 11, 19, 43, 23, 5, 2, 23, 3, 29, 5, 17, 3, 31, 29, 53, 29, 7, 13, 73, 3, 5, 43, 29, 17, 5, 37, 19, 11, 71, 7, 2, 43, 13, 19, 2, 59, 7, 29, 113, 13, 5, 11

Offset: 1

Author

J. M. Bergot and Robert Israel, Aug 08 2022

Keywords

• nonn

Examples

			a(2) = 19 because 19 is the start of the 2*2+1 = 5 consecutive primes 19, 23, 29, 31, 37 with 19*23 + 23*29 + 29*31 + 31*37 + 37*19 = 3853 prime, and no earlier 5-tuple of consecutive primes works.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A070934, A356471, A356475.

Programs

Maple

f:= proc(m) local P,x,i,n;
  n:= 2*m+1;
  P:= Vector(n,ithprime);
do
   x:= add(P[i]*P[i+1],i=1..n-1)+P[n]*P[1];
   if isprime(x) then return P[1] fi;
   P[1..n-1]:= P[2..n];
   P[n]:= nextprime(P[n]);
od
end proc:
map(f, [$1..100]);

Python

from sympy import isprime, nextprime, prime, primerange
def a(n):
    p = list(primerange(1, prime(2*n+1)+1))
    while True:
        if isprime(sum(p[i]*p[i+1] for i in range(len(p)-1))+p[-1]*p[0]):
            return p[0]
        p = p[1:] + [nextprime(p[-1])]
print([a(n) for n in range(1, 83)]) # Michael S. Branicky, Aug 08 2022