# [seqfan] Re: Middle digit in square numbers

M. F. Hasler seqfan at hasler.fr
Fri Dec 16 22:26:51 CET 2016

```Lars' & Andrew's idea can obviously be generalised to any base b.
So there's also:
Smallest number > 1 whose square's (cube's,...) middle digit in base n is
d, for d = 0,1,2,3...
(Without the " > 1 " it becomes trivial for d=0 and 1, and for all other d
the number is anyway > 1.)

This and the initial idea also make sense for d >= b (the base), with
"digit" replaced by "digits".
Then, when d has an even number of digits, the square also must have an
even number of digits.

When d is a square, then the smallest x whose square has d as middle
digit(s) is sqrt(d).
Thus, alternatively to " > 1 ", one could also impose " > sqrt( d ) "
to make it non-trivial for all d.

- Maximilian

Le 12 déc. 2016 17:13, "Andrew Weimholt" <andrew.weimholt at gmail.com> a
écrit :

> Numbers k, such that the middle binary digit of k^2 is 0
>
> 0, 2, 4, 5, 8, 9, 10, 16, 17, 18, 19, 22, 32, 33, 34, 35,
> 36, 37, 40, 41, 44, 64, 65, 66, 67, 68, 69, 70, 71, 76, 77, 80,
> 81, 84, 85, 87, 90, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138,
>
> Numbers k, such that the middle binary digit of k^2 is 1
>
> 1, 11, 20, 21, 38, 39, 42, 43, 45, 72, 73, 74, 75, 78, 79, 82,
> 83, 86, 88, 89, 140, 141, 142, 143, 148, 149, 150, 154, 155, 158, 159, 162,
> 163, 166, 167, 169, 170, 172, 173, 175, 178, 180, 181, 272, 273, 274, 275,
> 276,
>
> Andrew
>
>
> On Mon, Dec 12, 2016 at 10:28 AM, Charles Greathouse <
> charles.greathouse at case.edu> wrote:
>
> > Lars, on the contribution page
> > https://oeis.org/Submit.html
> > there is a link
> > https://oeis.org/edit/allocate
> > to allocate some A-numbers, which in my experience are always sequential.
> >
> > Charles Greathouse
> > Case Western Reserve University
> >
> > On Mon, Dec 12, 2016 at 5:18 AM, Lars Blomberg <larsl.blomberg at comhem.se
> >
> > wrote:
> >
> > > OK.
> > >
> > > So 10 A-numbers will be needed.
> > > It would facilitate greatly if a sequence like Axxxxx0 .. Axxxxx9 could
> > be
> > > allocated for this.
> > > Is that possible, if so, how?
> > > Regards,
> > > Lars Blomberg
> > >
> > > -----Ursprungligt meddelande-----
> > > Från: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] För
> > israel at math.ubc.ca
> > > Skickat: den 12 december 2016 08:39
> > > Till: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
> > > Ämne: [seqfan] Re: Middle digit in square numbers
> > >
> > > Not at all too simple.  Please do contribute them.
> > >
> > > Cheers,
> > > Robert
> > >
> > > On Dec 11 2016, Lars Blomberg 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
> > > >
> > > >
> > > >
> > > >
> > > >--
> > > >Seqfan Mailing list - http://list.seqfan.eu/
> > > >
> > > >
> > >
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> > >
> > > --
> > > Seqfan Mailing list - http://list.seqfan.eu/
> > >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>

```