A364091 a(n) is the first prime p such that the longest sequence of primes p = p_1, p_2, ... with |p_{k+1} - 2*p_k| = 1 has length n.

13, 7, 11, 5, 3, 2, 16651, 15514861, 85864769, 26089808579, 665043081119, 554688278429, 758083947856951, 95405042230542329, 69257563144280941

Offset: 1

Author

Robert Israel, Jul 04 2023

Keywords

• nonn
• more

Comments

a(n) = A000040(k) where A263879(k) = n is the first appearance of n in A263879.

Examples

			a(4) = 5 because 5, 2*5 + 1 = 11, 2*11 + 1 = 23, 2*23 + 1 = 47 is a sequence of primes of length 4 while 2*47 - 1 = 93 and 2*47 + 1 = 95 are not primes, and 5 is the smallest prime that works.
		

Crossrefs

Cf. A000040, A263879.

Programs

Maple

M:= 10: # for a(1) .. a(N)
f:= proc(n) option remember; local x;
  if n mod 3 = 1 then x:= 2*n-1 else x:= 2*n+1 fi;
  if isprime(x) then 1 + procname(x) else 1 fi;
end proc:
f(2):= 6: f(3):= 5:
V:= Vector(M):
p:= 1: count:= 0:
for k from 1 while count < M do
  p:= nextprime(p);
  v:= f(p);
  if v <= M and V[v] = 0 then V[v]:= p; count:= count+1; fi
od:
convert(V,list);

Python

from sympy import isprime, nextprime
def A364091(n):
    if 5 <= n <= 6: return 8-n
    q = 5
    while True:
        p, c = q, 1
        while isprime(p:=(p<<1)+(-1 if p%3==1 else 1)):
            c += 1
            if c > n:
                break
        if c == n:
            return q
        q = nextprime(q) # Chai Wah Wu, Jul 07 2023

Formula

a(n) = min(A005602(n), A005603(n)) for n >= 7.