[seqfan] Re: Adding digits by couples

[Jack Brennen]

I believe the smallest starting number which seems to
grow forever is 1496, which ends up at 1891 after
four iterations:

1496 -> 51315 -> 6446 -> 10810 -> 1891 -> ...



On 6/15/2012 11:26 AM, Jack Brennen wrote:
> After just 46 iterations, 1891 ends up with
> a number with 115236 digits; I think it's a
> safe conjecture that it grows forever.
>
> There is a pattern to it.  At the 19th iteration, the number
> begins with:
>
> 18168554419171361118161213169411181614106781186648741118161412106774461411111055
>
>
> And at the 20th iteration, the number begins with:
>
> 99714131098510108849722997733447151352299775516131592914121012151152299775533161
>
>
> Afterward, the first digits cycle through those two strings apparently
> ad infinitum.
>
>
>
> On 6/15/2012 10:57 AM, Eric Angelini wrote:
>> Hello Seqfans (I hope this idea is not old hat),
>>
>> We change integer abcde...mn (with a,b,c,d,e,..m,n being
>> digits) into integer a+b|b+c|c+d|d+e|e+...+m|m+n (where
>> the symbol < | > means < concatenate with >. Then we iterate.
>>
>> So 2012 gives 2+0|0+1|1+2 which is 213.
>>      213 gives 2+1|1+3 which is 34.
>>   And 34 gives 3+4 = 7 (which we call "end result")
>>
>> 1) What integers are divisible by their end result?
>>
>> 2) 991 loops -- as 992, 993, 994, 995, 996, 997, 998, 999
>>     (ex. 991-1810-991...); which integers > 999 also enter
>>     in a loop?
>>
>> 3) which is the smallest integer growing for ever (if such
>>     an integer exists)? I don't know the fate of 1891, for
>>     example: 1891-91710-10881-18169-99715-181686-9971414...
>>
>>   Besides, is there an end pattern here: 1991
>>                                        101810
>>                                         11991
>>                                       2101810
>>                                        311991
>>                                      42101810
>>                                       6311991
>>                                     942101810
>> Best,                                    ...
>> É.
>>
>>
>>
>>
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>>
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>>
>>
>>
>
>