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A382055 a(n) = least positive integer m such that when m*(m+1) is written in base n, it does not contain the digits 0 or n-1 and contains every single digit from 1 to n-2 exactly once, or 0 if no such number exists.

%I A382055 #17 Mar 17 2025 22:15:22
%S A382055 0,2,6,19,0,420,924,3672,0,78880,431493,2173950,0,71583429,436726936,
%T A382055 2750336517,0,120521201887,833996387274,5932255141224,0,
%U A382055 324116744376715,2483526997445916,19463766853506024,0,1274294107710603710,10627079743009611713,90335862784009245081,0
%N A382055 a(n) = least positive integer m such that when m*(m+1) is written in base n, it does not contain the digits 0 or n-1 and contains every single digit from 1 to n-2 exactly once, or 0 if no such number exists.
%F A382055 a(n) = 0 if n == 3 (mod 4). For a proof of this see A382054 for the proof of a similar result which also holds in this case.
%F A382055 Conjecture: a(n) = 0 if and only if n == 3 (mod 4).
%e A382055 a(9) = 924. 924*925 = 854700 which is 1542376 in base 9.
%o A382055 (Python)
%o A382055 from itertools import count
%o A382055 from math import isqrt
%o A382055 from sympy.ntheory import digits
%o A382055 def A382055(n):
%o A382055     k, l, d= (n*(n-1)>>1)%(n-1), (-n**(n - 1) + (n - 1)**2*(n**(n - 2)*(1 - n) + n**(n - 1)) + 1)//(n - 1)**2, tuple(range(1,n-1))
%o A382055     clist = [i for i in range(n-1) if i*(i+1)%(n-1)==k]
%o A382055     if len(clist) == 0:
%o A382055         return 0
%o A382055     s = (-n**2 + n + n**n - (n - 1)**3 - 1)//(n*(n - 1)**2)
%o A382055     s = isqrt((s<<2)+1)-1>>1
%o A382055     s += n-1-s%(n-1)
%o A382055     if s%(n-1) <= max(clist):
%o A382055         s -= n-1
%o A382055     for a in count(s,n-1):
%o A382055         if a*(a+1)>l:
%o A382055             break
%o A382055         for c in clist:
%o A382055             m = a+c
%o A382055             if m*(m+1)>l:
%o A382055                 break
%o A382055             if tuple(sorted(digits(m*(m+1),n)[1:]))==d:
%o A382055                 return m
%o A382055     return 0 # _Chai Wah Wu_, Mar 17 2025
%Y A382055 Cf. A381266, A382050, A382054.
%K A382055 nonn,base
%O A382055 3,2
%A A382055 _Chai Wah Wu_, Mar 13 2025