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

[Jonathan Post]

Here's a 4-initial-digit (base) analogue of A131573 Numbers whose
squares start with 3 identical digits.

Numbers whose squares start with 4 identical digits.

2357, 2582, 3334, 4714, 5774, 6667, 8165, 8819, 9428, 10541, 10542,
10543, 10544, 10545, 14907, 14908, 14909, 18257, 18258, 18259, 21081,
21082, 21083, 23570, 23571, 25819, 25820, 27888, 27889, 29813, 29814,
31622, 33332, 33333, 33334, 33335, 33336, 33337, 33338, 33339, 33340,
33341, 33342, 33343, 33344, 33345, 33346, 47139, 47140, 47141, 47142,
47143, 47144, 47145, 47146, 47147, 47148

Example: 2357^2 = 5555449, which I like because 2357 is the
concatenation of the first 4 primes.

Mathematica:

Select[Range[10, 50000], Length[Union[Take[IntegerDigits[#^2], 4]]] == 1 &]

======================

Analogue next:

Numbers whose squares start with 5 identical digits.

10541, 33334, 47141, 57735, 66667, 105409, 105410, 105411, 105412,
105413, 149071, 149072, 149073, 182574, 182575, 182576, 210818,
210819, 235702, 235703, 258198, 258199, 278886, 278887, 298141,
298142, 316227, 333332, 333333, 333334, 333335, 333336, 333337,
333338, 333339, 333340, 333341, 333342, 333343, 333344, 333345,
333346, 471403, 471404, 471405, 471406, 471407, 471408, 471409,
471410, 471411, 47141

Example: 10541^2 = 111112681.

Mathematica:

Select[Range[100, 500000], Length[Union[Take[IntegerDigits[#^2], 5]]] == 1 &]

======================

Analogue next:

Numbers whose squares start with 6 identical digits.

57735, 333334, 471405, 577350, 666667, 745356, 881917, 942809,
1054093, 1054094, 1054095, 1054096, 1490712, 1490713, 1490714,
1825741, 1825742, 1825743, 2108185, 2108186, 2357022, 2357023,
2581988, 2581989, 2788866, 2788867, 2981423, 2981424, 3162277,
3333332, 3333333, 3333334, 3333335, 3333336, 3333337, 3333338,
3333339, 3333340, 3333341, 3333342, 3333343, 3333344, 3333345,
3333346, 4714043, 4714044, 4714045, 4714046, 4714047, 4714048,
4714049, 4714050, 4714051, 4714052, 471405

Example: 57735^2 = 3333330225

Mathematica:

Select[Range[1000, 5000000],
          Length[Union[Take[IntegerDigits[#^2], 6]]] == 1 &]

======================

I know that there are people who hate "base" sequences, and OEIS is on
vacation anyway.  Are these already "less" interesting?  Is it better
to give a derived sequence by building the array A[k,n] = n-th
positive integer whose square (base 10) begins with k identical
digits.

Give it by antidiagonals, give A[n,n] the main diagonal?

Also, I'm not sure about the minima of my sequences, did I start too high?