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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.

%I A236383 #19 Jun 25 2019 01:33:42
%S A236383 428,453,465,381,369,358,917,421,394,452,704,716,442,833,323,380,347,
%T A236383 697,8376,449,3994,407,439,431,4770,6961,391,336,3533,4277,7915,36332,
%U A236383 7705,4487,3323,8869,8942,3250,4560,7632,90951,7988,4204,3606,8586,72774
%N 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.
%C A236383 Conjecture: a(n) exists for all numbers n.
%C A236383 a(1) = A030467(1).
%C A236383 The same problem with the concatenation of x + n instead of x + n^2 is difficult.
%C A236383 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
%H A236383 Michel Lagneau, <a href="/A236383/b236383.txt">Table of n, a(n) for n = 1..245</a>
%e A236383 a(11) = 704 because 704^2 = 495616 is the concatenation of 495 and 616, and 616 - 495 = 121 = 11^2.
%p A236383 for n from 1 to 47 do:
%p A236383    ii:=0:
%p A236383       for k from 1 to 10^7 while(ii=0)do :
%p A236383          x:=convert(k^2,base,10):n1:=nops(x):
%p A236383          if irem(n1,2)=0
%p A236383            then
%p A236383            s:=sum('x[i]*10^(i-1) ', 'i'=1..n1/2):
%p A236383            z:=convert(s,base,10):
%p A236383            s1:=sum('x[j]*10^(j-n1/2-1) ', 'j'=n1/2+1..n1):
%p A236383             if s-s1 = n^2
%p A236383             then
%p A236383             ii:=1:printf(`%d, `,k):
%p A236383             else
%p A236383             fi:
%p A236383          fi:
%p A236383        od:
%p A236383    od:
%Y A236383 Cf. A030467.
%K A236383 nonn,base
%O A236383 1,1
%A A236383 _Michel Lagneau_, Jan 24 2014
%E A236383 Definition corrected by _Giovanni Resta_, Jun 24 2019