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

Jonathan Post jvospost3 at gmail.com
Thu Aug 30 02:53:02 CEST 2007

```Here's a 4-initial-digit (base) analogue of A131573 Numbers whose

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:

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:

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?

```