[seqfan] Re: Positive integers m such that sum of decimal digits of (16^n - 1) equals 6*n.

[Vladimir Shevelev]

P.S. Since the phenomenon of large
jumps in sequences occurs quite often,
I suggest to introduce a unit of a big
jump in a sequence: 1 kangaroo (1 kan) 
which means 
a new difference equals 1000 times 
the maximal from the previous differences
in their absolute values.

I wish everyone a wonderful day!

Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Vladimir Shevelev [shevelev at exchange.bgu.ac.il]
Sent: 29 November 2017 18:58
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Positive integers m such that sum of decimal digits of (16^n - 1) equals 6*n.

We with Peter Moses did one more experiment.
It is clear that digit sum of (16^n-1) in base 4 is always 6*n.

Therefore, I ask Peter to continue sequence
1,1,3,3,4,6,6,7,... for which a(n)=S(n)/5, where S(n) is Sum
of digits of 16^n-1 in base 6. He obtained

1, 1, 3, 3, 4, 6, 6, 6, 9, 6, 8, 14, 8, 10, 12, 11, 15, 15, 15, 15, 20, 19, 19, 18, 18, 23, 24, 26, 24, 23, 25, 32, 26, 23, 26, 28, 31, 29, 27, 31, 32, 37, 34, 31, 35, 34, 33, 38, 35, 36, 43, 44, 43, 45, 45, 43, 47, 45, 48, 43, 44, 53, 53, 41, 48, 55, 55, 54,...

We see that the fixed points are 1,3,6,9,32,... For them sum of digits of 16^n-1 in base 6 is 5*n. I ask Peter  to continue this sequence,

but up to 50000 he did not obtain any term, i.e., in the sequence of fixed points we have a very
large jump.

It is interesting, if to consider different bases b for the same expression, say, 16^n-1,
what is the smallest number N(b) for which we have a very large jump (in comparing with
previous differences). Here there is no a strong definition for "very large jump" (for example
more than 1000 or 10000 times more than every previous difference, but it is characterised
as "explosion" or "resonance", etc.)

Best regards,
Vladimir
________________________________________
From: SeqFan [seqfan-bounces at list.seqfan.eu] on behalf of Max Alekseyev [maxale at gmail.com]
Sent: 29 November 2017 04:41
To: Sequence Fanatics Discussion list
Subject: [seqfan] Re: Positive integers m such that sum of decimal digits of (16^n - 1) equals 6*n.

It's worth to quote my old message regarding the phenomenon of having so
many terms in this sequence.
Below S(n) denotes the sum of decimal digits of n.

---------- Forwarded message ----------
From: Max A. <maxale at gmail.com>
Date: Sun, Oct 29, 2006 at 5:52 PM
Subject: Re: Sum of decimal digits of 16^n - 1 = 6*n

[...]

The reason is that S(n) == n (mod 9).
One can easily check that 16^n - 1 == 6*n (mod 9) holds for all n
while 16^n - 1 == 5*n (mod 9) holds only for n == 0 (mod 9).

btw, these are some other values of a and k such that the sum of
digits of a^n - 1 equals k*n for n=1,2,...,m:

m=9:
a=53941, k=21
a=539410, k=30

m=8:
a=136195 k=24
a=963055, k=27

m=7:
a=18811, k=18
a=18901, k=18
a=27244, k=18
a=40771, k=18

E.g., the sum of digits of 53941^n - 1 equals 21*n for the following n:
1, 2, 3, 4, 5, 6, 7, 8, 9, 13, 21, 31, 40, 48, 59, 67, 98, 114, 119,
130, 140, 148, 151, 156, 174, 186, 190, 242, 280, 335, 361, 367, 414,
418, 421, 425, 451, 511, 567, 569, 653, 789, 898, 1027, 1321, 1340,
2277, 2416

Max

On Tue, Nov 28, 2017 at 8:59 PM, <israel at math.ubc.ca> wrote:

> I think it's extremely unlikely that anything close to this could be
> proved in the current state of the art. There just aren't the tools to
> handle such questions. I would be surprised if we could prove there isn't
> an n such that all digits of 16^n-1 are >= 5.
>
> Cheers,
> Robert
>
>
> On Nov 28 2017, Iain Fox wrote:
>
> Hello all, I recently submitted a conjecture to this sequence (A165722)
>> that for n greater than 223 the sum of digits of 16^n - 1 will always be
>> less than 6*n, meaning that the sequence is finite. I have tested this up
>> to n=10^6 so it seems likely. I, however, have no clue about how to go
>> about proving this. If anyone has any ideas it would be appreciated.
>>
>> Full text format of sequence:
>>
>> %I A165722 %S A165722 1,2,3,4,5,6,7,10,12,13,14,17,18,23,37,43,46,60,119,183,223
>> %N A165722 Positive integers n such that the sum of decimal digits of (16^n
>> - 1) equals 6*n. %C A165722 No other terms below 10^4. %C A165722
>> Conjecture: For n > 223, digsum(16^n - 1) < 6*n. This would mean that no
>> further terms exist in the sequence. - _Iain Fox_, Nov 22 2017 %C A165722
>> No other terms below 10^6. - _Iain Fox_, Nov 25 2017 %F A165722
>> A007953(16^n - 1) = A008588(n). - _Iain Fox_, Nov 22 2017 %e A165722 For
>> n=1, 16-1 is 15 with sum of digits 6, so 1 is a term. %e A165722 For n=2,
>> 16^2-1 is 255 with sum of digits 12, so 2 is a term. %t A165722
>> Select[Range[250],6#==Total[IntegerDigits[16^#-1]]&] (* _Harvey P.
>> Dale_, Nov 13 2012 *) %o A165722 (PARI) is(n) = 6*n == sumdigits(16^n-1) \\
>> _Iain Fox_, Nov 24 2017 %K A165722 base,more,nonn,changed %O A165722 1,2 %A
>> A165722 _Max Alekseyev_, Sep 24 2009
>>
>> --
>> 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/