A236383 Smallest k such that k^2 is a concatenation of two numbers x and y where y = x + n^2 and x and y have the same number of digits.

428, 453, 465, 381, 369, 358, 917, 421, 394, 452, 704, 716, 442, 833, 323, 380, 347, 697, 8376, 449, 3994, 407, 439, 431, 4770, 6961, 391, 336, 3533, 4277, 7915, 36332, 7705, 4487, 3323, 8869, 8942, 3250, 4560, 7632, 90951, 7988, 4204, 3606, 8586, 72774

Offset: 1

Author

Michel Lagneau, Jan 24 2014

Keywords

• nonn
• base

Comments

Conjecture: a(n) exists for all numbers n.

a(1) = A030467(1).

The same problem with the concatenation of x + n instead of x + n^2 is difficult.

The corresponding sequence with x + n instead of x + n^2 starts with 36363636364, 428, 8874, 5, 310, 7, 39 for n = 0,...,6, and a(7) > 10^70, if it exists. - Giovanni Resta, Jun 24 2019

Examples

			a(11) = 704 because 704^2 = 495616 is the concatenation of 495 and 616, and 616 - 495 = 121 = 11^2.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..245

Crossrefs

Cf. A030467.

Programs

Maple

for n from 1 to 47 do:
   ii:=0:
      for k from 1 to 10^7 while(ii=0)do :
         x:=convert(k^2,base,10):n1:=nops(x):
         if irem(n1,2)=0
           then
           s:=sum('x[i]*10^(i-1) ', 'i'=1..n1/2):
           z:=convert(s,base,10):
           s1:=sum('x[j]*10^(j-n1/2-1) ', 'j'=n1/2+1..n1):
            if s-s1 = n^2
            then
            ii:=1:printf(`%d, `,k):
            else
            fi:
         fi:
       od:
   od:

Extensions

Definition corrected by Giovanni Resta, Jun 24 2019