A345223 a(n) is the smallest k >= 0 such that the decimal concatenation 1 (n times) || k || 1 (n times) is a prime, or -1 if no such k exists.

0, 3, 4, 8, 10, 8, 5, 21, 1, 6, 1, 116, 23, 6, 73, 24, 16, 62, 3, 10, 19, 53, 61, 58, 191, 9, 265, 12, 133, 86, 141, 4, 7, 39, 193, 31, 51, 13, 31, 6, 31, 53, 287, 139, 4, 239, 187, 25, 18, 144, 31, 38, 93, 86, 27, 30, 16, 24, 6, 356, 50, 91, 395, 117, 217, 61

Offset: 1

Author

Felix Fröhlich, Jun 11 2021

Keywords

• nonn
• base
• less

Comments

a(n) = 0 only for n = 1, since A138148(1) = 101 is the only prime in A138148.

a(n) = 1 iff n is of the form (A004023(i)-1)/2 for some i >= 1.

No term equals 2, see second comment in A258372.

Examples

			For n = 3: 1110111, 1111111, 1112111 and 1113111 are all composite, while 1114111 is prime, so the smallest number that can be inserted between strings of three ones so that the concatenation is prime is 4. Therefore a(3) = 4.
		

Crossrefs

Cf. A004023, A088281, A138148, A258372.

Programs

Mathematica

Table[Module[{k=0},While[!PrimeQ[FromDigits[Flatten[Join[{PadRight[ {},n,1],IntegerDigits[ k],PadRight[{},n,1]}]]]],k++];k],{n,70}] (* Harvey P. Dale, Jun 03 2024 *)

PARI

eva(n) = subst(Pol(n), x, 10)
a(n) = my(v=vector(n, t, 1), d, w=[]); for(k=0, oo, d=digits(k); w=concat(v, d); w=concat(w, v); if(ispseudoprime(eva(w)), return(k)))

Python

from sympy import isprime
def a(n, d=1):
    k, bread = 0, str(d)*n
    while not isprime(int(bread + str(k) + bread)): k += 1
    return k
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Jun 11 2021