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

21, 9045, 890445, 88904445, 8889044445, 888890444445, 88888904444445, 8888889044444445, 888888890444444445, 88888888904444444445, 8888888889044444444445, 973609801090486804900545, 88888888888904444444444445, 8888888888889044444444444445, 931379640537060465689820268530

Offset: 1

Author

Robert Israel, Feb 04 2025

Keywords

• nonn
• base

Comments

a(n) is the largest triangular number of the form (2*10^n+1)*x where 10^(n-1) <= x < 10^n / 2.

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).

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..100
• Mathematics StackExchange, What is the largest 6-digit number ABCDEF where ABC = 2DEF which can be expressed as the sum of the first n natural numbers?

Crossrefs

Cf. A000217, A226742.

Programs

Maple

g:= proc(d) local a,b,n, ymax,x,y;
      ymax:= -1;
      for a in numtheory:-divisors(2*(2*10^d+1)) do
        b:= 2*(2*10^d+1)/a;
        if igcd(a,b)>1 then next fi;
        n:= chrem([0,-1],[a,b]);
        x:= n*(n+1)/2;
        y:= x/(2*10^d+1);
        if y < 10^(d-1) or 2*y >= 10^d then next fi;
        if y > ymax then ymax:= y fi
      od;
      (2*10^d+1)*ymax
end proc:
map(g, [$1..25]);