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

1,1

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

Links

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

Example

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

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:

Crossrefs

Cf. A030467.

Sequence in context: A251147 A095811 A235217 * A224668 A343536 A030467

Adjacent sequences: A236380 A236381 A236382 * A236384 A236385 A236386

Keyword

nonn,base

Author

Michel Lagneau, Jan 24 2014

Extensions

Definition corrected by Giovanni Resta, Jun 24 2019

Status

approved