[seqfan] Re: Potential improvements to existing OEIS Taxicab sequences

Sat Jan 23 14:17:50 CET 2021

Dear Maximilian,

Many thanks for responding to my query. I will address your main point
below directly.

“Did you notice that A003825 already has 34204 terms listed? So I'm not
sure that it needs completion, unless you want to say that there are
missing terms within the list given so far? If you think so, please say
this clearly (and if possible give the first missing term(s)), it would be
very important. Since the b-file is from Ray Chandler I would tend to think
it's correct, but I didn't check.”

Yes, I did notice that A003825 already has 34,204 terms listed,

Is it complete or incomplete? I note it does state “primitive solutions”.

But I believe just as A001235 has 30,000 terms as a *complete* list, I
believe it should be relatively straightforward to produce a comparable
*complete* list for A003825 based on the current list.

So for example, for *T(3,1) =* *87,539,319*

When T(3,k) is expressed in the form of the simple power equation, T(3,1)1
= A1 x (a1)13, where, T(3,1)1 = 87,539,319, and (a1)1= 167, A1 could be
calculated. Then by using the equation T(3,1)i= A1 x (a1)i3, where, i = 1,
2, 3, 4, …. ∞, setting  (a1)1 = i x a1, and by substituting the calculated
value for A1 the first 14 values of the infinite sequence were determined
and are given below. This infinite sequence is one which can be expressed
as the sum of two positive cubes in three different ways. Other than the
first term in the sequence (i.e., 87,539,319), none of the other terms are
given in the published ordered sequence for T(3,k). This is the first
sub-set of T(3,k) and the first 14 terms of the sequence are given below.

*First ordered sub-set of the ordered sequence T(3,k), which can be
expressed as the sum of two positive cubes in three different ways,
T(3,1)i, given for the first 14 terms of the infinite sequence.*

*i*

*T(3,1)i*

*(a1)i*

*(b1)i*

*(c1)i*

*(d1)i*

*(e1)i*

*(f1)i*

*1*

*87,539,319*

*167*

*436*

*228*

*423*

*255*

*414*

2

700,314,552

334

872

456

846

510

828

3

2,363,561,613

501

1308

684

1269

765

1242

4

5,602,516,416

668

1744

912

1692

1020

1656

5

10,942,414,875

835

2180

1140

2115

1275

2070

6

18,908,492,904

1002

2616

1368

2538

1530

2484

7

30,025,986,417

1169

3052

1596

2961

1785

2898

8

44,820,131,328

1336

3488

1824

3384

2040

3312

9

63,816,163,551

1503

3924

2052

3807

2295

3726

10

87,539,319,000

1670

4360

2280

4230

2550

4140

11

116,514,833,589

1837

4796

2508

4653

2805

4554

12

151,267,943,232

2004

5232

2736

5076

3060

4968

13

192,323,883,843

2171

5668

2964

5499

3315

5382

14

240,207,891,336

2338

6104

3192

5922

3570

5796

Let’s refer to the current list as *T*(3,k)*

One might expect the cardinality of T(3,k) to be greater than that of
T*(3,k) because the infinite number of mutually exclusive sub-sets, T(3,k)i,
where each value of each infinite sequence, T(3,k)i, does *not* have a 1:1
correspondence with the ordered, infinite sequence, T*(3,k).

This is not the case with the list for T(2,k).

The subsets of T(2,k) are related to T(2,k), by the simple power
equation, T(2,k)i
= Ai x (ak)i3

T(2,k), can:

(1)    Be partitioned into mutually exclusive, infinite, ordered sub-sets

(2)    With each ordered subset (i) forming an infinite ordered sequence,
expressed as Ai x (ak)i3

(3)    The infinite, ordered, mutually exclusive sub-sets of sequences,
T(2,k)i, have a 1:1 correspondence with T(2,k).

For further details, please refer to 3 full manuscripts (non-peered
reviewed) available here:

https://www.researchgate.net/profile/Laurence_Lacey/publications

Any feedback on any of these manuscripts would be gratefully received. I
hope my responses above addresses you main points of feedback.

Best regards, Larry.

On Fri, Jan 22, 2021 at 11:06 PM M. F. Hasler <seqfan at hasler.fr> wrote:

> On Tue, Jan 19, 2021 at 8:44 AM larry lacey <larryflacey at gmail.com> wrote:
>
> > (3)    There is a simple method of mine that can be used to complete the
> > sequences oeis.org/A003825 & oeis.org/A003826
> >
>
> Hi Larry,
>
> did you notice that  A003825 already has 34204 terms listed?
> So I'm not sure that it needs completion,
> unless you want to say that there are missing terms within the list given
> so far?
> If you think so, please say this clearly (and if possible give the first
> missing term(s)), it would be very important.
> Since the b-file is from Ray Chandler I would tend to think it's correct,
> but I didn't check.
>
> OTOH, the Mmca code given by Orlovsky in ois.org/A018787 ("all solutions",
> i.e., not only non-primitive) looks indeed faulty to me!
> The list that code would produce should indeed be incomplete in the second
> half, if I'm not wrong.
> Specifically, if one lists solutions as large as  m^3+n^3 with  m<n<2000,
> then one actually needs to go up to n = 2000 * 2^(1/3) ~ 2500 to be sure
> one does not miss those which have 2nd and 3rd representations with much
> smaller m.
> But again, the b-file (here 100 000 terms !) was computed by Ray Chandler,
> not by Orlovsky, so there's hope it's correct.
> Please be more explicit if you have found an error in the database.
>
> Thank you,
>
> - Maximilian
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>