Further terms:
14670238462896430 < a(6) < 14670469667698570;
1.88655928870547380*10^22 < a(7) < 1.8865698003644555*10^22;
1.5845871199286*10^29 < a(8) < 1.5845909238805*10^29;
3.7023896360635*10^36 < a(9) < 3.7023941021398*10^36;
1.0075615552422*10^45 < a(10) < 1.0075622026833*10^45;
1.3480809599483*10^53 < a(11) < 1.3480814844466*10^53;
3.9618565460983*10^62 < a(12) < 3.9618574860993*10^62;
7.8648507615953*10^71 < a(13) < 7.864851991241*10^71;
4.7945106758325*10^81 < a(14) < 4.7945111864185*10^81;
1.0953005932169*10^92 < a(15) < 1.0953006746693*10^92;
8.3149001821943*10^148 < a(20) < 8.3149003278317*10^148.
All terms are even, since a(n) + 1 is always an odd prime number.
The numbers a(n) + 1 and a(n) + 2 are zero containing numbers (in base n).
The numbers between a(n) and p := min( k > a(n) + 1 | k is prime) are zero containing numbers, i.e., a(n) + j is a zero containing number for 0 < j < p - a(n).
For numbers m > a(10) = 1.00756...*10^45, we have pi(m) > A324161(m) [= number of zerofree numbers <= m]; in general, the ratio A324161(m) to pi(m) is O(log(n)*n^d), where d := 1 - 1/(1 - log_10(9)) = -0.0457..., and thus tends to 0 for m --> infinity. Consequently, the share of primes <= m which have no '0'-digit become significantly smaller as m rises beyond that bound a(10). For m = 10^100, the share is not greater than 0.000688, for m = 10^1000, the share cannot exceed 4.52757*10^(-43). Conversely, the share of primes which contain a '0'-digit tends to 1 as m grows to infinity (cf. A011540).
Conjecture: a(n) can be represented in the form a(n) = k*n^m + j, where j < (n^(m+1)-1)/(n-1) - n^m, and m > 1, 0 < k < n.
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