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A371511 a(n) is the smallest prime such that its representation in base n contains each of the digits 0,1,...,n-2 at least once and does not contain the digit n-1.

%I A371511 #14 Apr 13 2024 23:17:33
%S A371511 3,73,683,8521,123323,2140069,43720693,1012356487,26411157737,
%T A371511 749149003087,23459877380431,798411310382011,29471615863458281,
%U A371511 1158045600182881261,48851274656431280857,2193475267557861578041,104737172422274885174411,5257403213296398892278377
%N A371511 a(n) is the smallest prime such that its representation in base n contains each of the digits 0,1,...,n-2 at least once and does not contain the digit n-1.
%C A371511 Conjecture: for n>3, a(n) has digit sum 2+(n-2)(n-1)/2 if n is of the form 4k+3 and has digit sum 1+(n-2)(n-1)/2 otherwise.
%H A371511 Chai Wah Wu, <a href="/A371511/b371511.txt">Table of n, a(n) for n = 3..387</a>
%H A371511 Chai Wah Wu, <a href="https://arxiv.org/abs/2403.20304">Pandigital and penholodigital numbers</a>, arXiv:2403.20304 [math.GM], 2024.
%F A371511 For n>3, a(n) >= (n^(n-1)-n)/(n-1)^2 + n^(n-1). If n = 4k+3 for k>0, then a(n) >= (n^(n-1)-n)/(n-1)^2 + n^(n-1) + n^(n-3) .
%e A371511 The corresponding base-n representations are:
%e A371511 n   a(n) in base n
%e A371511 ------------------------
%e A371511 3   10
%e A371511 4   1021
%e A371511 5   10213
%e A371511 6   103241
%e A371511 7   1022354
%e A371511 8   10123645
%e A371511 9   101236457
%e A371511 10  1012356487
%e A371511 11  10223456798
%e A371511 12  10123459a867
%e A371511 13  1012345678a9b
%e A371511 14  1012345678c9ab
%e A371511 15  1022345678a9cdb
%e A371511 16  10123456789acbed
%o A371511 (Python)
%o A371511 from math import gcd
%o A371511 from sympy import nextprime
%o A371511 from sympy.ntheory import digits
%o A371511 def A371511(n):
%o A371511     m, j = n, 0
%o A371511     if n > 3:
%o A371511         for j in range(1,n-1):
%o A371511             if gcd((n*(n-1)>>1)+j,n-1) == 1:
%o A371511                  break
%o A371511     if j == 0:
%o A371511         for i in range(2,n-1):
%o A371511             m = n*m+i
%o A371511     elif j == 1:
%o A371511         for i in range(1,n-1):
%o A371511             m = n*m+i
%o A371511     else:
%o A371511         for i in range(2,1+j):
%o A371511             m = n*m+i
%o A371511         for i in range(j,n-1):
%o A371511             m = n*m+i
%o A371511     m -= 1
%o A371511     while True:
%o A371511         s = digits(m:=nextprime(m), n)[1:]
%o A371511         if n-1 not in s and len(set(s))==n-1:
%o A371511             return m
%Y A371511 Cf. A185122, A371194.
%K A371511 nonn,base
%O A371511 3,1
%A A371511 _Chai Wah Wu_, Apr 10 2024