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tech / sci.math / Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...

SubjectAuthor
* Squares with N 9's = 9, 3969, 29929, 1999396, ...henh...@gmail.com
`* Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...henh...@gmail.com
 `* Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...Timothy Golden
  `* Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...James Waldby
   `- Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...Timothy Golden

1
Squares with N 9's = 9, 3969, 29929, 1999396, ...

<9c5798e8-3d5c-4b2e-83bb-fe124516084en@googlegroups.com>

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Subject: Squares with N 9's = 9, 3969, 29929, 1999396, ...
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Sat, 18 Jun 2022 18:09 UTC

(2236 squared) ===== 4,999,696
(83666 squared) ===== 6999999556

https://oeis.org/A036516
Smallest square containing exactly n 9's.
9, 3969, 29929, 1999396, 299739969, 2909199969, 19299599929, 909995799969, 9499999990849, .........

i'm wondering if there's something interesting about these numbers.

1.-- How the numbers tend to be ( 19999... 29999... ) is a bit like Benford's law

2. -- In the following, where the 1st column has [0], the square is div. by 9.

Perhaps these (div. by 9) squares in this list will become more rare (as the numbers get bigger) ----- just a hunch/guess

0 - 9
0 - 3969
2 - 29929
1 - 1999396
0 - 299739969
0 - 2909199969
2 - 19299599929
0 - 909995799969
1 - 9499999990849
1 - 999999202999696
0 - 9969959993997969
1 - 90949999999997329
2 - 9199999971969929929
1 - 199999969997999200969
1 - 19997911999199999979409
2 - 295191999919994299999969
1 - 9599845999999199998999729
1 - 29999999992959999995699281
1 - 199922979699799999999939969

Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...

<49bb490b-6475-4274-b2ad-79c6c1b61446n@googlegroups.com>

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Subject: Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...
From: henha...@gmail.com (henh...@gmail.com)
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 by: henh...@gmail.com - Sat, 18 Jun 2022 18:19 UTC

> (2236 squared) ===== 4,999,696
> (83666 squared) ===== 6999999556
>
> https://oeis.org/A036516
> Smallest square containing exactly n 9's.
> 9, 3969, 29929, 1999396, 299739969, 2909199969, 19299599929, 909995799969, 9499999990849, .........
>
i'm wondering if there's something interesting about these numbers.
1.-- How the numbers tend to be ( 1999... 2999... ) is a bit like Benford's law
2. -- In the following, where the 1st column has [0], the square is div. by 9.
Perhaps these (div. by 9) squares in this list will become more rare (as the numbers get bigger)
----- just a hunch/guess

> 0 - 9
> 0 - 3969
> 2 - 29929
> 1 - 1999396
> 0 - 299739969
> 0 - 2909199969
> 2 - 19299599929
> 0 - 909995799969
> 1 - 9499999990849
> 1 - 999999202999696
> 0 - 9969959993997969
> 1 - 90949999999997329
> 2 - 9199999971969929929
> 1 - 199999969997999200969
> 1 - 19997911999199999979409
> 2 - 295191999919994299999969

295191999919994299999969 --- according to Google, this number shows up in exactly one Web page, and the following numbers don't show up at all ------ which is a bit surprising.

Maybe... even with fast computers, extending list with a few more numbers is pretty difficult.

> 1 - 9599845999999199998999729
> 1 - 29999999992959999995699281
> 1 - 199922979699799999999939969

Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...

<eff4bbc7-dcdd-4a42-9d17-8f206591948an@googlegroups.com>

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Subject: Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Sun, 19 Jun 2022 12:37 UTC

On Saturday, June 18, 2022 at 2:19:53 PM UTC-4, henh...@gmail.com wrote:
> > (2236 squared) ===== 4,999,696
> > (83666 squared) ===== 6999999556
> >
> > https://oeis.org/A036516
> > Smallest square containing exactly n 9's.
> > 9, 3969, 29929, 1999396, 299739969, 2909199969, 19299599929, 909995799969, 9499999990849, .........
> >
> i'm wondering if there's something interesting about these numbers.
> 1.-- How the numbers tend to be ( 1999... 2999... ) is a bit like Benford's law
> 2. -- In the following, where the 1st column has [0], the square is div. by 9.
>
> Perhaps these (div. by 9) squares in this list will become more rare (as the numbers get bigger)
> ----- just a hunch/guess
>
>
> > 0 - 9
> > 0 - 3969
> > 2 - 29929
> > 1 - 1999396
> > 0 - 299739969
> > 0 - 2909199969
> > 2 - 19299599929
> > 0 - 909995799969
> > 1 - 9499999990849
> > 1 - 999999202999696
> > 0 - 9969959993997969
> > 1 - 90949999999997329
> > 2 - 9199999971969929929
> > 1 - 199999969997999200969
> > 1 - 19997911999199999979409
> > 2 - 295191999919994299999969
> 295191999919994299999969 --- according to Google, this number shows up in exactly one Web page, and the following numbers don't show up at all ------ which is a bit surprising.
>
> Maybe... even with fast computers, extending list with a few more numbers is pretty difficult.
> > 1 - 9599845999999199998999729
> > 1 - 29999999992959999995699281
> > 1 - 199922979699799999999939969

x = 999...9
x*x = 999...9 000...0 1

Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...

<t8nu16$cfb$1@dont-email.me>

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From: j-wal...@no.no (James Waldby)
Newsgroups: sci.math
Subject: Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...
Date: Sun, 19 Jun 2022 19:39:18 -0000 (UTC)
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 by: James Waldby - Sun, 19 Jun 2022 19:39 UTC

Timothy Golden <timbandtech@gmail.com> wrote:
> On Saturday, June 18, 2022 at 2:19:53 PM UTC-4, henh...@gmail.com wrote:
>> > (2236 squared) ===== 4,999,696
>> > (83666 squared) ===== 6999999556
>> >

>> > https://oeis.org/A036516 Smallest square containing exactly n
>> > 9's. 9, 3969, 29929, 1999396, 299739969, 2909199969,
>> > 19299599929, 909995799969, 9499999990849, .........

>> i'm wondering if there's something interesting about these numbers.
>> 1.-- How the numbers tend to be ( 1999... 2999... ) is a bit like
>> Benford's law 2. -- In the following, where the 1st column has [0],
>> the square is div. by 9. Perhaps these (div. by 9) squares in this
>> list will become more rare (as the numbers get bigger) ----- just a
>> hunch/guess

>> > 0 - 9
>> > 0 - 3969
>> > 2 - 29929
>> > 1 - 1999396
>> > 0 - 299739969
>> > 0 - 2909199969
>> > 2 - 19299599929
>> > 0 - 909995799969
>> > 1 - 9499999990849
>> > 1 - 999999202999696
>> > 0 - 9969959993997969
>> > 1 - 90949999999997329
>> > 2 - 9199999971969929929
>> > 1 - 199999969997999200969
>> > 1 - 19997911999199999979409
>> > 2 - 295191999919994299999969
>> > 1 - 9599845999999199998999729
>> > 1 - 29999999992959999995699281
>> > 1 - 199922979699799999999939969
[...]

>> Maybe... even with fast computers, extending list with a few more
>> numbers is pretty difficult.

> x = 999...9
> x*x = 999...9 000...0 1

That representation of the multiplied result is incorrect. There is
an 8 digit in the square, between the 9's and 0's. (10^k -1)^2 =
10^(2k) - 2*10^k + 1, which gives k-1 9's in (10^k -1)^2's base 10
representation.

Note, the name of sequence A036516 is "Smallest square containing
exactly n 9's". Let A[j] be the jth entry. (10^k -1)^2 often is more
than k/4 orders of magnitude larger than A[k-1]. Eg, A[3]=1999396 and
(10^4 -1)^2 = 99980001, with ratio about 50. Eg, A[5]=299739969 while
(10^6 -1)^2 = 999998000001, with ratio about 3336.

Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...

<73bf84b7-9174-4ee0-8dd2-1558714c26b0n@googlegroups.com>

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Subject: Re: Squares with N 9's = 9, 3969, 29929, 1999396, ...
From: timbandt...@gmail.com (Timothy Golden)
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 by: Timothy Golden - Sun, 19 Jun 2022 21:24 UTC

On Sunday, June 19, 2022 at 3:39:27 PM UTC-4, James Waldby wrote:
> Timothy Golden <timba...@gmail.com> wrote:
> > On Saturday, June 18, 2022 at 2:19:53 PM UTC-4, henh...@gmail.com wrote:
> >> > (2236 squared) ===== 4,999,696
> >> > (83666 squared) ===== 6999999556
> >> >
>
> >> > https://oeis.org/A036516 Smallest square containing exactly n
> >> > 9's. 9, 3969, 29929, 1999396, 299739969, 2909199969,
> >> > 19299599929, 909995799969, 9499999990849, .........
>
> >> i'm wondering if there's something interesting about these numbers.
> >> 1.-- How the numbers tend to be ( 1999... 2999... ) is a bit like
> >> Benford's law 2. -- In the following, where the 1st column has [0],
> >> the square is div. by 9. Perhaps these (div. by 9) squares in this
> >> list will become more rare (as the numbers get bigger) ----- just a
> >> hunch/guess
>
> >> > 0 - 9
> >> > 0 - 3969
> >> > 2 - 29929
> >> > 1 - 1999396
> >> > 0 - 299739969
> >> > 0 - 2909199969
> >> > 2 - 19299599929
> >> > 0 - 909995799969
> >> > 1 - 9499999990849
> >> > 1 - 999999202999696
> >> > 0 - 9969959993997969
> >> > 1 - 90949999999997329
> >> > 2 - 9199999971969929929
> >> > 1 - 199999969997999200969
> >> > 1 - 19997911999199999979409
> >> > 2 - 295191999919994299999969
> >> > 1 - 9599845999999199998999729
> >> > 1 - 29999999992959999995699281
> >> > 1 - 199922979699799999999939969
> [...]
> >> Maybe... even with fast computers, extending list with a few more
> >> numbers is pretty difficult.
> > x = 999...9
> > x*x = 999...9 000...0 1
> That representation of the multiplied result is incorrect. There is
> an 8 digit in the square, between the 9's and 0's. (10^k -1)^2 =
> 10^(2k) - 2*10^k + 1, which gives k-1 9's in (10^k -1)^2's base 10
> representation.

Ahhh. Very good find. Yes, I missed that 8.
x*x = 999...9 8 000...0 1

Still it seems none will legitimate this form.
Of course nobody will refute it either.
At least you've worked it.
You are the first.

>
> Note, the name of sequence A036516 is "Smallest square containing
> exactly n 9's". Let A[j] be the jth entry. (10^k -1)^2 often is more
> than k/4 orders of magnitude larger than A[k-1]. Eg, A[3]=1999396 and
> (10^4 -1)^2 = 99980001, with ratio about 50. Eg, A[5]=299739969 while
> (10^6 -1)^2 = 999998000001, with ratio about 3336.

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