Numbers whose squares start with 4 identical digits, with 5, with 6...

[Jonathan Post]

Numbers whose squares start with 7 identical digits.

In[9]:= Select[Range[1000, 5000000],
          Length[Union[Take[IntegerDigits[#^2], 7]]] == 1 &]

Out[9]= {745356, 942809, 1490712, 1825742, 3333334, 4714045, 4714046}

Example:

745356^2 = 555555566736



Jonathan Post wrote:

> Numbers whose squares start with 7 identical digits.
>
> In[9]:= Select[Range[1000, 5000000],
>           Length[Union[Take[IntegerDigits[#^2], 7]]] == 1 &]
>
> Out[9]= {745356, 942809, 1490712, 1825742, 3333334, 4714045, 4714046}
>
> Example:
>
> 745356^2 = 555555566736

Here's an example of a more generic Mathematica program that  
calculates the first q terms of squares starting with n identical  
digits:

In[1]:=

n = 17; q = 10; t =
Table[{(10^n - 1)*i/9, (10^n - 1)*i/9 + 1}, {i, 1, 9}]; t =
Sqrt[Union[t, 10*t]]; k = 1; While[
k}]]];
Length[s] < q, k++]; Take[s, q]

Out[1]:=

{7453559924999299, 10540925533894598, 14907119849998598,  
33333333333333334, 47140452079103169, 57735026918962577,  
66666666666666667, 74535599249992990, 81649658092772603,  
88191710368819686}

In[2]:=

%^2

Out[2]:=

{55555555555555555733985150491401, 111111111111111115889741773581604,  
222222222222222222935940601965604,  
1111111111111111155555555555555556,  
2222222222222222288842087345842561,  
3333333333333333396736221926480929,  
4444444444444444488888888888888889,  
5555555555555555573398515049140100,  
6666666666666666622046303867395609, 7777777777777777716034080781138596}

It's a quick hack so I apologize if there's something wrong with it.  
I'm not totally happy with the 10^-1000 fudge-factor: it's just meant  
to reduce the number that it's part of by one, if it should ever turn  
out to be an exact integer.

Hans