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

1,2

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.

Links

Table of n, a(n) for n=1..66.

Example

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.

Prog

(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

Crossrefs

Cf. A004023, A088281, A138148, A258372.

Sequence in context: A113057 A253606 A258640 * A185726 A152520 A239976

Adjacent sequences: A345220 A345221 A345222 * A345224 A345225 A345226

Keyword

nonn,base,less

Author

Felix Fröhlich, Jun 11 2021

Status

approved