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

1,1

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).

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

Example

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

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]);

Prog

(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

Crossrefs

Cf. A004023, A068160.

Sequence in context: A280838 A140105 A265715 * A178496 A186433 A342076

Adjacent sequences: A366413 A366414 A366415 * A366417 A366418 A366419

Keyword

nonn,base,look

Author

Robert Israel, Oct 10 2023

Status

approved