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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.
3
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
1,1
LINKS
EXAMPLE
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.
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]);
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Aug 08 2022
STATUS
approved