[seqfan] Re: Sequence A055629

[David Wilson]

On 10/8/2011 3:47 AM, mathstutoring wrote:
> Morning David
> I have just been looking at your sequence A055629 and noticed that 
> a(6) and a(7) are the same
> Surely that is not possible with this type of sequence
> If you translate the data into consecutive runs you get 1, 2, 3, 4, 5, 
> 7, 7
> Now whilst you could argue that 7 consecutive runs includes 6, it is 
> my understanding that such sequences are not constructed like this and 
> that a(6) should represent the first run of exactly six 
> consecutive happy numbers and not at least six. Even if a(6)>a(7).
> What do you think
> Ant King (UK)

Let me start by saying this is not "my sequence". It is true that I 
authored the sequence some years ago.
However, since a(6) and a(7) are too large to be found by brute force 
search, even with aid of a computer,
and because I don't know of a clever way to compute them, I can safely 
say that I did not include a(6) and
a(7) in my original submission; these must have been added by someone 
else at a later date.

At any rate, you are correct that I could argue that a run of 7 numbers 
includes a run of 6 numbers. In fact,
I could, and will, argue that this is the correct mathematical 
interpretation. Whereas a non-mathematician
might construe "run of 6 numbers" to mean "run of exactly 6 numbers", a 
mathematician would not make
this unstated assumption. The sequence title is "Beginning of first run 
of n consecutive happy numbers", not
"Beginning of first run of exactly n consecutive happy numbers", so I 
think a(6) = a(7), if correct, is consistent
with the title as a mathematical description.

When you say "it is my understanding that such sequences are not 
constructed like this", I take it to mean
that you believe the OEIS uses the term "run of" with the consistent 
meaning of "run of exactly".  This is not
the case.  To its credit, many OEIS titles explicitly indicate "run of 
at least" (A064708, A077647, A078143)
or "run of exactly" (A049523, A055623, A064709), but some titles that 
say only "run of" when they mean
either "run of exactly" (A006558) or "run of at least" (A069561). This 
may be an issue of interest to the OEIS
editors, so I will copy them on this reply.

Regarding A055629, I think the best thing to do is to clarify its 
definition to

      Beginning of first run of at least n consecutive happy numbers

I agree that this is in some sense unsatisfying, since we would like 
A055629 to tell us where the first run of
exactly 6 happy numbers starts. To fix this, we might want to change its 
definition to

      Beginning of first run of exactly n consecutive happy numbers

If we do this, the current value of a(6) becomes unknown. OEIS policy 
then requires us to truncate
A055629 prior to a(6), so a(6) and a(7) must be removed.  We could 
perhaps add a comments that a(6) is
unknown (and may not exist) and that a(7) = 7899999999999959999999996.