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A228191 a(n) is the smallest number m such that the m-th triangular number ends in n zeros.

%I A228191 #21 Jul 26 2022 01:36:04
%S A228191 4,24,624,9375,90624,890624,7109375,12890624,212890624,1787109375,
%T A228191 81787109375,81787109375,81787109375,59918212890624,259918212890624,
%U A228191 3740081787109375,56259918212890624,256259918212890624,7743740081787109375,7743740081787109375
%N A228191 a(n) is the smallest number m such that the m-th triangular number ends in n zeros.
%C A228191 The numbers are often one less than the equivalent entry in A067270. - _R. J. Mathar_, Aug 27 2013
%C A228191 Because T(n)-n = T(n-1), the terms of this sequence would always be one less than the corresponding entry in A067270 if that sequence included repetitions with leading zeros. For instance, T(81787109376) ends in ...081787109376, so both 81787109376 and 081787109376 could be included. - _Robert Dawson_, Aug 31 2018
%H A228191 Chai Wah Wu, <a href="/A228191/b228191.txt">Table of n, a(n) for n = 1..1000</a> (terms 1..100 from Giovanni Resta)
%e A228191 a(2)=24 because 24 is the smallest number such that 24th triangular number i.e. 300 ends in 2 '0's.
%t A228191 a = {}; m = 1; Do[b = n*(n + 1)/2; If[Mod[b, 10^m] == 0, m = m + 1; AppendTo[a, n]], {n, 1, 1000000000}]; a
%t A228191 sol[k_, u_] := Block[{x}, Min[x /. List@ToRules[Reduce[Mod[x + u, 2*2^k] == 0 && Mod[x + 1 - u, 5^k] == 0 && x > 0, {x}, Integers] /. C[1] -> 0]]]; a[n_] := Min[sol[n, 0], sol[n, 1]]; a /@ Range[20] (* _Giovanni Resta_, Aug 15 2013 *)
%o A228191 (Python)
%o A228191 from sympy.ntheory.modular import crt
%o A228191 def A228191(n): return int(min(crt(m:=(1<<(n+1),5**n),(0,-1))[0], crt(m,(-1,0))[0])) # _Chai Wah Wu_, Jul 25 2022
%Y A228191 Cf. A000217, A228188.
%K A228191 nonn,base
%O A228191 1,1
%A A228191 _Shyam Sunder Gupta_, Aug 15 2013
%E A228191 a(10)-a(20) from _Giovanni Resta_, Aug 15 2013