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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.

%I A345223 #13 Jun 03 2024 19:11:35
%S A345223 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 A345223 265,12,133,86,141,4,7,39,193,31,51,13,31,6,31,53,287,139,4,239,187,
%U A345223 25,18,144,31,38,93,86,27,30,16,24,6,356,50,91,395,117,217,61
%N 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.
%C A345223 a(n) = 0 only for n = 1, since A138148(1) = 101 is the only prime in A138148.
%C A345223 a(n) = 1 iff n is of the form (A004023(i)-1)/2 for some i >= 1.
%C A345223 No term equals 2, see second comment in A258372.
%e A345223 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 A345223 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 A345223 (PARI) eva(n) = subst(Pol(n), x, 10)
%o A345223 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 A345223 (Python)
%o A345223 from sympy import isprime
%o A345223 def a(n, d=1):
%o A345223     k, bread = 0, str(d)*n
%o A345223     while not isprime(int(bread + str(k) + bread)): k += 1
%o A345223     return k
%o A345223 print([a(n) for n in range(1, 67)]) # _Michael S. Branicky_, Jun 11 2021
%Y A345223 Cf. A004023, A088281, A138148, A258372.
%K A345223 nonn,base,less
%O A345223 1,2
%A A345223 _Felix Fröhlich_, Jun 11 2021