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A380792 a(n) is the largest triangular number that is the concatenation of two n-digit numbers 2*x and x.

%I A380792 #19 Feb 05 2025 22:12:15
%S A380792 21,9045,890445,88904445,8889044445,888890444445,88888904444445,
%T A380792 8888889044444445,888888890444444445,88888888904444444445,
%U A380792 8888888889044444444445,973609801090486804900545,88888888888904444444444445,8888888888889044444444444445,931379640537060465689820268530
%N A380792 a(n) is the largest triangular number that is the concatenation of two n-digit numbers 2*x and x.
%C A380792 a(n) is the largest triangular number of the form (2*10^n+1)*x where 10^(n-1) <= x < 10^n / 2.
%C A380792 For n >= 2, (2*10^n+1)*(4*(10^n-1)/9+1) = t * (t+1)/2 where t = (4*10^n + 2)/3, so this is a triangular number of that form.  Thus a(n) >= (2*10^n+1)*(4*(10^n-1)/9+1).
%H A380792 Robert Israel, <a href="/A380792/b380792.txt">Table of n, a(n) for n = 1..100</a>
%H A380792 Mathematics StackExchange, <a href="https://math.stackexchange.com/questions/5031550/what-is-the-largest-6-digit-number-abcdef-where-abc-2def-which-can-be-expres">What is the largest 6-digit number ABCDEF where ABC = 2DEF which can be expressed as the sum of the first n natural numbers?</a>
%e A380792 a(3) = 890445 because 890445 = 1334 * 1335/2 is a triangular number and is the concatenation of the two 3-digit numbers 890 = 2*445 and 445, and no larger number works.
%p A380792 g:= proc(d) local a,b,n, ymax,x,y;
%p A380792       ymax:= -1;
%p A380792       for a in numtheory:-divisors(2*(2*10^d+1)) do
%p A380792         b:= 2*(2*10^d+1)/a;
%p A380792         if igcd(a,b)>1 then next fi;
%p A380792         n:= chrem([0,-1],[a,b]);
%p A380792         x:= n*(n+1)/2;
%p A380792         y:= x/(2*10^d+1);
%p A380792         if y < 10^(d-1) or 2*y >= 10^d then next fi;
%p A380792         if y > ymax then ymax:= y fi
%p A380792       od;
%p A380792       (2*10^d+1)*ymax
%p A380792 end proc:
%p A380792 map(g, [$1..25]);
%Y A380792 Cf. A000217, A226742.
%K A380792 nonn,base
%O A380792 1,1
%A A380792 _Robert Israel_, Feb 04 2025