%I #13 Jun 03 2024 19:11:35
%S 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,
%T 265,12,133,86,141,4,7,39,193,31,51,13,31,6,31,53,287,139,4,239,187,
%U 25,18,144,31,38,93,86,27,30,16,24,6,356,50,91,395,117,217,61
%N 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.
%C a(n) = 0 only for n = 1, since A138148(1) = 101 is the only prime in A138148.
%C a(n) = 1 iff n is of the form (A004023(i)-1)/2 for some i >= 1.
%C No term equals 2, see second comment in A258372.
%e 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.
%t 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 *)
%o (PARI) eva(n) = subst(Pol(n), x, 10)
%o 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)))
%o (Python)
%o from sympy import isprime
%o def a(n, d=1):
%o k, bread = 0, str(d)*n
%o while not isprime(int(bread + str(k) + bread)): k += 1
%o return k
%o print([a(n) for n in range(1, 67)]) # _Michael S. Branicky_, Jun 11 2021
%Y Cf. A004023, A088281, A138148, A258372.
%K nonn,base,less
%O 1,2
%A _Felix Fröhlich_, Jun 11 2021