A366416 a(n) is the first prime that starts and ends with at least n 1's (in base 10).

11, 11, 1114111, 111181111, 111110611111, 1111118111111, 111111151111111, 111111110911111111, 1111111111111111111, 1111111111111111111, 1111111111111111111, 1111111111111111111, 1111111111111111111, 1111111111111111111, 1111111111111111111, 1111111111111111111, 1111111111111111111

Offset: 1

Author

Robert Israel, Oct 10 2023

Keywords

• nonn
• base
• look

Comments

The initial and final strings of 1's are allowed to overlap.

If k is in A004023 and (k+1)/2 <= j <= k, then a(j) = (10^k-1)/9 (unless it is (10^i-1)/9 for some i < k where i is in A004023 and (i+1)/2 <= j <= i).

Examples

			a(3) = 1114111 which is prime and starts and ends with 3 1's.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..495
• S. Dutta et al, Infinitely many primes of the form 11...1 (k times) ... 11...1 (k times), Mathematics StackExchange

Crossrefs

Cf. A004023, A068160.

Programs

Maple

f:= proc(n) local x,s,d;
  for d from n to 2*n-1 do
     if isprime((10^d-1)/9) then return (10^d-1)/9 fi
  od;
  s:= (10^n-1)/9;
  for d from n do
    for x from 10^d*s + s by 10^n to 10^d*(s+1) do
      if isprime(x) then return x fi
  od od
end proc:
map(f, [$1..20]);

Python

from gmpy2 import is_prime
def a(n):
    t = (10**n-1)//9
    for d in range(n, 2*n):
        if is_prime(t): return t
        t = 10*t + 1
    suffix = (10**n-1)//9
    d = 2*n
    while True:
        prefix = 10**(d-n)*suffix
        for mid in range(0,10**(d-n),10**n):
            t = prefix + mid + suffix
            if is_prime(t): return t
        d += 1
print([a(n) for n in range(1,18)]) # Michael S. Branicky, Oct 10 2023