Further terms:
4.1645275173242*10^15 < a(6) < 4.164601237609*10^15,
1.0163657136*10^22 < a(7) < 1.0163715977928*10^22,
8.4513797224747*10^28 < a(8) < 8.4514006058085*10^28,
1.959502408617*10^36 < a(9) < 1.9595048275153*10^36,
1.0953002073198*10^44 < a(10) < 1.0953009588121*10^44,
1.3480809599483*10^53 < a(11) < 1.3480814844466*10^53,
3.540916347013*10^61 < a(12) < 3.5409172310273*10^61,
2.080341784427*10^71 < a(13) < 2.0803421176765*10^71,
2.4843833393543*10^81 < a(14) < 2.4843836067277*10^81,
5.6615671922884*10^91 < a(15) < 5.6615676172791*10^91,
2.1556069128839*10^148 < a(20) < 2.1556069510899*10^148.
a(n) is always a prime number.
For n > 2, all terms are odd.
All terms a(n) are zero-containing numbers (in base n), a(n) - 1 is also a zero-containing number (in base n).
The numbers between p := max( k < a(n) | k is prime) and a(n) + 1 are zero-containing numbers, i.e., a(n) + 1 - j is a zero-containing number for 0 < j < a(n) + 1 - p.
From the equality A324164(5) = A324165(5) we can conclude that a(5) and A324155(5) + 1 are proximate primes. Same is true for a(11): a(11) and A324155(11) + 1 are proximate primes.
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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