[seqfan] help naming/describing sequences for bounds of Goldbach's Comet, was: what happened to my A342302, its been replaced ???

[peter lawrence]

All,
     I agree that the encyclopedia itself maybe isn’t the right place to work out the details, 
so lets do that here in seqfan instead.

Goldbach's Comet (e.g.  https://en.wikipedia.org/wiki/Goldbach%27s_comet <https://en.wikipedia.org/wiki/Goldbach's_comet>,  and lots of google images,  etc)
has obvious upper and lower bounds, at least visually, so the natural question is what are those bounds.

I’m using A002372 for Goldbach counts (order dependent sums) because they can be computed as a convolution,
and because there’s something obvious about when the value is even verse odd that is much less easy to state
when using A002375 (unordered sums).

note that A002372 isn't indexed by N, rather it's indexed by N/2 because odd numbers aren’t generally
the sum of two primes (in general only the upper half of a twin prime pair is), and the factor of two 
makes it awkward to relate that sequence to ones discussed here where I talk about what N achieves a 
bound rather than what N/2.

it would be nice to have approximation formulas, 
and it would also be nice to have sequences that could be found in OEIS,
I’d like to work on both, but for this email I’ll stick with sequences.

for the upper bound sequence one might tabulate where a high point is first achieved, as in

1 is first achieved at N = 6,    6 = {3+3} = 1 way
2 is first achieved at N = 8,    8 = {3+5, 5+3} = 2 ways
3 is first achieved at N = 10,   10 = {3+7, 5+5, 7+3} = 3 ways
4 is first achieved at N = 16,   16 = {3+13, 5+11, 11+5, 13+3} = 4 ways
5 is first achieved at N = 22,   22 = {3+19, 5+17, 11+11, 17+5, 19+3} = 5 ways
etc...

so   6,8,10,16,22,...   maybe should be in OEIS ?

well not so fast, continuing we find that the first and last N for each possible count 1..20 are
count first last
  1     6     6
  2     8    12
  3    10    38
  4    16    68
  5    22    62
  6    24   128
  7    34   122
  8    36   152
  9    74   158      --- 9 isn’t achieved until N=74, but a couple higher counts (10,120 are achieved earlier
 10    48   188
 11   106   166      --- ditto
 12    60   332
 13   178   398      --- and so it goes for all odd counts since an odd count only
 14    78   272
 15   142   362      --- occurs for N = 2 x Prime which are more rare than N = 2 x Composite
 16    84   368
 17   202   458
 18    90   488
 19   358   542
 20   114   632

so the strict upper bounds (points on the (top side of the) convex hull of Goldbach’s Comet)
don’t include all possible counts, example ‘9’ above.

also note that for just even counts the first occurrence isn’t always steadily increasing
 30   234   908
 32   246  1112
 34   288   968
 36   240  1412     — the count 36 occurs earlier than 32, and 34
 38   210  1178     — the count 38 occurs earlier than 30, 32, 34, and 36
 40   324  1448

so not all even counts (which in general occur earlier than odd counts) are on the (top 
side of the) convex hull either.

similarly the “last occurrence” sequence contains points not on the (bottom side of the) convex
hull of Goldbach’s Comet.

[One might ask, what is the best way to put points on the convex hull of Goldbach’s Comet into OEIS,
 if you have a suggestion feel free to comment, those are what I consider the true upper and lower
 bounds, but those are pairs of integers. Yes they are integers, yes they can be sequenced, but they
 are pairs. Would you create two (related) sequences ?  I’m not proposing anything on that topic,
 instead...]


but even though the above sequences aren’t the lower and upper bounds I was looking for,
and one might call them longitudinal studies of Goldbach’s Comet instead,
they still seem to be of sufficient mathematical interest to be in OEIS, because

There are some obvious conjectures
1.  every count is the Goldbach count for some N 
    equivalently every number occurs in A002372 (there is a first occurrence)
2.  every count is the Goldbach count for only finitely many N,
    equivalently there is a last occurrence in A002372 for each number


so I hope folks on this list can help with viable descriptions for these sequences
such that they will be accepted into the OEIS


what would you name these sequences

(I consider them well defined, IMHO, but) how would you describe these sequences

if you don’t think they are well defined please state why



Peter A Lawrence



> On Apr 18, 2021, at 6:51 PM, Neil Sloane <njasloane at gmail.com> wrote:
> 
> Peter Lawrence, 
> Following the advice of some senior editors - but based primarily on my own views - your sequence was rejected by me on April 2 2021 as "interesting but not ready for the OEIS".  You would have a received a copy of this decision since you were the author.
> 
> Furthermore, this was also recorded in the webpage on the OEIS Wiki called Deleted Sequences.
> 
> The reasons are documented in the Pink Box discussions of the sequence, which you can see by going to the "history" tab of A342302.
> 
> Best regards
> Neil 
> 
> Neil J. A. Sloane, President, OEIS Foundation.
> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ.
> Phone: 732 828 6098; home page: http://NeilSloane.com <http://neilsloane.com/>
> Email: njasloane at gmail.com <mailto:njasloane at gmail.com>
>