# [seqfan] Re: Middle digit in square numbers

Jeremy Gardiner jeremy.gardiner at btinternet.com
Mon Dec 12 09:46:22 CET 2016

```Consider also cubes:

0: 0, 30, 40, 42, 100, 101, 115, 116, 123, 126, 135, 163, 164, 171, 199,
1: 1, 6, 8, 23, 44, 45, 102, 106, 110, 114, 117, 121, 137, 148, 152, 153,
2: 5, 9, 103, 113, 133, 146, 151, 154, 165, 180, 198,
3: 29, 34, 39, 46, 118, 125, 141, 142, 155, 161, 170,
4: 7, 104, 112, 140, 143, 158, 166, 186, 188, 195,
5: 26, 43, 107, 109, 119, 122, 136, 139, 144, 150, 177, 179, 197,
6: 22, 25, 27, 36, 37, 124, 129, 134, 147, 156, 160, 169,
7: 31, 32, 105, 111, 128, 130, 149, 167, 173, 191, 192,
8: 2, 24, 35, 38, 120, 127, 131, 138, 145, 172, 174, 182, 183,
9: 28, 33, 41, 108, 132, 157, 159, 175, 178, 181, 184, 187, 190, 193, 196,

Which brings to mind these others:

Numbers whose squares have an odd number of digits:

0, 1, 2, 3, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27, 28, 29, 30, 31, 100,

The corresponding squares:

0, 1, 4, 9, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484,
529, 576, 625, 676, 729, 784, 841, 900, 961, 10000,

Numbers whose cubes have an odd number of digits:

0, 1, 2, 5, 6, 7, 8, 9, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 100,

The corresponding cubes:

0, 1, 8, 125, 216, 343, 512, 729, 10648, 12167, 13824, 15625, 17576, 19683,
21952, 24389, 27000, 29791, 32768, 35937, 39304, 42875,

Jeremy Gardiner

On 12/12/16 06:34, "Lars Blomberg" <larsl.blomberg at comhem.se> wrote:

>

Hello,

Numbers k such that the middle digit of k^2 is d.

Obviously k^2
> must have an odd number of digits.

Examples for d=4: 2^2 = 4, 12^2 = 144,
> 21^2 = 441, 29^2 = 841, 102 = 10404,
...

Start of sequences for d =
> 0..9

0: 10, 20, 30, 100, 105, 138, 145, .

1: 1, 110, 119, 123, 127, 131,
> 142, .

2: 11, 15, 18, 23, 25, 27, 101, 106,.

3: 111, 124, 128, 139, 146,
> 156, 177, .

4: 2, 12, 21, 29, 102, 107, 116, 120,.

5: 16, 112, 150, 163,
> 166, 169, 172,.

6: 13, 19, 31, 103, 108, 117, 121, .

7: 24, 26, 113, 137,
> 144, 154, 181,.

8: 17, 22, 28, 104, 109, 122, 126, .

9: 3, 14, 114, 118,
> 130, 134, 148, .

It was a surprise to me that none of these sequences are
> in OEIS.

Maybe they are too simple to be included?

/Lars Blomberg

>

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```