[seqfan] Re: Product of digits touching the comma = first difference

[David Wilson]

Among the d-digit numbers, every number less than 10^d-81 has a successor.
If a d-digit number is farther than 81 from a (d+1)-digit number, it has 
a successor.

For d >= 2, the d-digit numbers without successors all start with digit 
9. We
will henceforth assume d >= 2.

If your starting number does not start with digit 9, your sequence will 
continue
until you reach a number ending in 0 (after which the sequence becomes 
constant),
or you reach a numbers starting with 9.

Consider a number which precedes a number starting with 9. Let its last 
digit be d.
The difference between the two numbers is 9d, so number starting with 9 must
also end in 0. Thus if we ever reach a number starting with 9 after the 
first element,
it will end in 0, and the sequence will be constant thereafter.

So, if a sequence starts with a d-digit number not in the "bad range" 
[10^d-81, 10^d-1],
it will continue until it reaches a number ending with 0 (on or before 
reaching
a number starting with 9) and will become constant.

Some of the starting numbers in this "bad range" will have no successors 
and end
immediately,  others will have successors staring with 9, and therefore 
ending in 0,
and will become constant at the second element, others will jump the 
boundary to
the (d+1)-digit numbers then go on by the previous argument to reach a 
number ending
in 0 and become constant.

>
> -----Original Message-----
> From: seqfan-bounces at list.seqfan.eu [mailto:seqfan-bounces at list.seqfan.eu]
> On Behalf Of Eric Angelini
> Sent: 08 November 2011 12:08 PM
> To: Sequence Fanatics Discussion list
> Cc: Jean-Marc Falcoz; Alexandre Wajnberg
> Subject: [seqfan] Product of digits touching the comma = first difference
>
>
> Hello SeqFans,
>
> S = 96, 102, 104, 108, 116, 122, 124, 128, 136,
>      142, 144, 148, 156, 162, 164, 168, 176, 182,
>      184, 188, 196, 208, 216, 228, 244, 252, 256,
>      268, 284, 292, 296, 314, ...
>
> The first difference between two adjacent integers is the product of the
> digits touching the comma:
>
> S = 96, 102, 104, 108, 116, 122, 124, 128, 136,
> dif    6    2    4    8    6    2    4    8
>
> Is 96 the smallest integer producing an infinite such sequence? (96 has no
> predecessor.)
>
> Note that 9, for instance, has 8 possible succes- sors (18,27,36... 81) --
> 18 itself having 5 pos- sible successors (58,66,74,82,90).
>
> 384 has two possible successors (396 and 400).
>
> Integers ending with zero loop on themselves:
> 10,10,10,...
>
> If 96 doesn't produce an infinite sequence, is there an integer which does?
>