Why isn't 0 in A034710?

[David Wilson]

----- Original Message ----- 
From: "Max Alekseyev" <maxale at gmail.com>
To: "Jonathan Post" <jvospost3 at gmail.com>
Cc: "Sequence Fans" <seqfan at ext.jussieu.fr>; "jonathan post" 
<jvospost2 at yahoo.com>; "Erich Friedman" <erich.friedman at stetson.edu>; "Larry 
Reeves" <larryr at acm.org>
Sent: Friday, September 21, 2007 11:45 PM
Subject: Re: Why isn't 0 in A034710?


> On 9/21/07, Jonathan Post <jvospost3 at gmail.com> wrote:
>
>> Why isn't 0 in A034710  Numbers for which the sum of digits equals the
>> product of digits?
>
> I think this sequence assume positive integers.
>> That is, why not correct A034710 to 0, 1, 2, 3, 4,
>> 5, 6, 7, 8, 9, 22, 123, 132, 213, 231, 312, 321, 1124, ...
>> with offset 0,3?
>
> Why the offset is 0? Even having 0 in the sequence shouldn't change
> the offset to 0. As many other sequences representing certain subsets
> of the integers, it should have the offset 1.

Mentally, I tend to classify most sequences as function-type sequences 
(where the values are a function of the indices) or set-type sequences 
(where the elements satisfy a property). set-type sequences have distinct 
elements, and if they have no negative elements, they are increasing.

When I encounter such an increasing set-type sequence of nonnegative 
integers, I usually consider 0 as a sort of optional element (since 
sometimes the inclusion of 0 in the sequence is arguable, or else the reader 
is interested only in positive elements of the set). If 0 is in the 
sequence, I index starting at 0 (so that a(0) = 0), otherwise, I start at 1. 
That way, nonnegative (positive) indices map to nonnegative (positive) 
values, and a(1) is always the first positive element. This works quite 
nicely, and avoids having to adjust the index and potentially invalidate 
formulae if we later decide to add or omit 0.

>> 0 would also thus be in A061672 Smallest number formed by a set
>> of digits whose product = sum of the digits?
>
> A061672 also contains positive integers.
>
> Zero is a very special number, its leading digit is 0 that is not
> allowed for other numbers. If we try to fit the number zero into this
> rule, it should contain no digits at all (the "empty" number) with the
> sum of digits equal 0 and the product of digits equal 1.
>
> But I would better vote for excluding zero from the aforementioned
> sequences, as it is now.

And if we decide 0 belongs in the sequence, we could set a(0) = 0 without 
disturbing existing indexing scheme.

> Regards,
> Max
>
>
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I will chnage the definition of A034710 to
which I think is the intent of this sequence.

Neil



David Wilson wrote to say that A113461 is effectively equal to A061558

I'm not so sure.  It may not be true.

I've added a new sequence A130791 to clarify things.
An equivalent question is, is A113460 the same (apart from the
leading term) as A130791?

One approach that might settle this would be
to extend all these sequences, and that would be a good
thing to do in any case.

On the other hand there may be a simple proof 
that these pairs of sequences really are the same?

Neil