# [seqfan] Re: Are these really the same?

Veikko Pohjola veikko at nordem.fi
Sun Jan 8 09:29:33 CET 2012

```Thanks a lot gentlemen. I need to admit that I intuitively knew that this
would be the case but was perhaps too lazy to continue searching. The next
counterexamples to be found at this level would be 31 and 40. So we can say
that with this appproach, by adding the nesting of digit sums, it is
possible to produce sequences whose coincidence with A156638 reaches further
and further in an asymptotic way, but the sequences are never 'really the
same'.
Veikko

----- Original Message -----
From: "David Wilson" <davidwwilson at comcast.net>
To: "Sequence Fanatics Discussion list" <seqfan at list.seqfan.eu>
Sent: Sunday, January 08, 2012 3:58 AM
Subject: [seqfan] Re: Are these really the same?

> On 1/7/2012 6:20 PM, Veikko Pohjola wrote:
>> I produced a sequence identical to A156638 as follows: Numbers n such
>> that the digit sum of the digit sum of the square of the digit sum of n^2
>> = 4. Can it be proved or disproved that these are really the same?
>>
>> Veikko Pohjola
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
> A156638 consists of all numbers == 4 or 5 (mod 9). A fairly
> straightforward argument modulo 9 shows
> that all the numbers you describe are in A156638. However, there are
> numbers in A156638 that are
> not in your sequence. For example, let
>
> n =
> 96019407833469901546749865786386112278884519841349077509706112489402136224573901
> 82990862369255038178669819071254543866242079874381931885715723232715356395266377
> 02691450018096418683273171301219560970962323567670685363467717396610425927598342
> 139
>
> n == 4 (mod 9), so n is in A156638.
>
> Letting d(x) be the digit sum of x, you can also show that
>
>            d(n^2) = 2167
>     ==> d(n^2)^2 = 4695889
>     ==> d(d(n^2)^2) = 49
>     ==> d(d(d(n^2)^2)) = 13
>
> since 13 is not 4, n is not one of your numbers. Hence your sequence is
> not A156638.
>
> The counterexample n has 243 digits. n is certainly not the smallest
> counterexample, but my gut tells me
> that the smallest counterexample likely has > 100 digits, leastwise, your
> sequence and A156638 coincide for
> a VERY VERY long time before a difference is encountered.
>
> If your sequence description had yet another "the digit sum of" in front
> of it, then the n above would also be
> in your sequence. The sequence would still not be the same as A156638,
> however, if we could write digits on
> subatomic particles, the known universe would not contain enough particles
> to write down the first
> counterexample.
>
>
>
>
>
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>

```