[seqfan] Re: Help with a(n) and a cumulative sum

[israel at math.ubc.ca]

There are infinite sequences such that a(n) doesn't share any digit with 
the cumulative sum. For example: 1, 6, 4, 66, 34, 666, 334, ... with 
cumulative sums 1, 7, 11, 77, 111, 777, 1111, ... So your sequence A309151 
can indeed be infinite.

Cheers,
Robert

On Jul 14 2019, Éric Angelini wrote:

> Hello SeqFans, Here is the draft of a strange sequence: 
> https://oeis.org/draft/A309151 It says: Lexicographically earliest 
> sequence of different terms starting with a(1) = 1 such that a(n) doesn't 
> share any digit with the cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) 
> + a(n). And in the Comments section: As this sequence needs a lot of 
> backtracking, we don't guarantee the accuracy of the last 79 integers of 
> the 1079-term b-file. Indeed, the problem comes from the fact that some 
> cumulative sums quickly block the extension of the sequence. This is the 
> case with 10 (or any other sum ending in zero). But this is the case too 
> with 301 after 
> 1,2,3,5,4,6,7,8,9,10,11,12,14,22,20,23,24,30,13,15,16,18,28. So, we 
> bumped quite often in a "bad sum" (on the average, the sequence was 
> extended by 100 terms for every backtrack). To make a prior list of "bad 
> sums" is difficult (meaning impossible, I guess): 258002 is such a "bad 
> sum" if you have previously used {1,3,4,6,7,9,39,49,69,79} else 258002 + 
> 79 = 258081 would be ok. So my questions are: could the sequence be 
> infinite? Could a list of "bad sum numbers" be easely defined and used? 
> Post-scriptum: I am working, together with Jean-Marc, on the variants: 
> "Lexicographically [...] such that a(n) shares exacly k digit with the 
> cumulative sum a(1) + a(2) + a(3) + ... + a(n-1) + a(n)", with k = 1, 2, 
> 3, ... Best, É.
>
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