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A371512 a(n) is the smallest prime such that its representation in base n contains each of the digits 1,...,n-2 at least once and does not contain the digit 0 nor the digit n-1.

%I A371512 #11 Apr 13 2024 23:17:42
%S A371512 13,37,163,1861,22481,304949,5455573,112345687,2831681057,68057976031,
%T A371512 1953952652167,61390449569437,2224884906436873,77181689614101181,
%U A371512 3052505832274232281,129003238915759600789,6090208982148446231753,276667213296398892309917,13944042713948404997174231
%N A371512 a(n) is the smallest prime such that its representation in base n contains each of the digits 1,...,n-2 at least once and does not contain the digit 0 nor the digit n-1.
%C A371512 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 A371512 Chai Wah Wu, <a href="/A371512/b371512.txt">Table of n, a(n) for n = 3..388</a>
%H A371512 Chai Wah Wu, <a href="https://arxiv.org/abs/2403.20304">Pandigital and penholodigital numbers</a>, arXiv:2403.20304 [math.GM], 2024.
%F A371512 For n>=3, a(n) >= (n^(n-1)-n)/(n-1)^2 + n^(n-2). If n = 4k+3 for k>0, then a(n) >= (n^(n-1)-n)/(n-1)^2 + n^(n-2) + n^(n-3) .
%e A371512 The corresponding base-n representations are:
%e A371512 n   a(n) in base n
%e A371512 ------------------------
%e A371512 3   111
%e A371512 4   211
%e A371512 5   1123
%e A371512 6   12341
%e A371512 7   122354
%e A371512 8   1123465
%e A371512 9   11234567
%e A371512 10  112345687
%e A371512 11  1223456987
%e A371512 12  1123458a967
%e A371512 13  112345678ba9
%e A371512 14  11234567a8bc9
%e A371512 15  122345678acb9d
%e A371512 16  1123456789ceabd
%o A371512 (Python)
%o A371512 from math import gcd
%o A371512 from sympy import nextprime
%o A371512 from sympy.ntheory import digits
%o A371512 def A371512(n):
%o A371512     m, j = 1, 0
%o A371512     if n > 3:
%o A371512         for j in range(1,n-1):
%o A371512             if gcd((n*(n-1)>>1)+j,n-1) == 1:
%o A371512                  break
%o A371512     if j == 0:
%o A371512         for i in range(2,n-1):
%o A371512             m = n*m+i
%o A371512     elif j == 1:
%o A371512         for i in range(1,n-1):
%o A371512             m = n*m+i
%o A371512     else:
%o A371512         for i in range(2,1+j):
%o A371512             m = n*m+i
%o A371512         for i in range(j,n-1):
%o A371512             m = n*m+i
%o A371512     m -= 1
%o A371512     while True:
%o A371512         s = digits(m:=nextprime(m), n)[1:]
%o A371512         if (not (0 in s or n-1 in s)) and len(set(s))==n-2:
%o A371512             return m
%Y A371512 Cf. A185122, A371194.
%K A371512 nonn,base
%O A371512 3,1
%A A371512 _Chai Wah Wu_, Apr 10 2024