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

Thu Jan 28 12:33:39 CET 2021

Dear Maximilian & other SeqFans,

Maybe the best way forward would be to retain A003825, with its 34,204
“primitive solutions”, but to compliment it with a complete list using the
method I have outlined previously? Both lists could have their uses to
those interested in these sequences.

Furthermore, would it be useful to have the following description of the
complete set of Taxicab numbers?

The complete set of infinite taxicab numbers, T(n,k)i, has the following
characteristics:

1)      The sequence of positive integer values, T(n,k), approaches
infinity, as the positive integer, k, approaches infinity, for any given
positive integer value of n.

2)      Each positive integer value of T(n,k), for any given value of n and
k, respectively, has an infinite sub-set of positive integer sequences,
T(n,k)i

where, T(n,k)I = Ai x (ak)i3

with ak being the smallest cubic term, satisfying the condition: (ak)3 + (bk
)3 = T(n,k)

3)      The infinite complete set of taxicab numbers, T(n,k)i, comprises
three distinct infinities:

-          as n approaches infinity, the taxicab “hierarchy” number,
approaches infinity

-          as k approaches infinity, the k positive integer sequence value,
for any given taxicab “hierarchy” number, n, approaches infinity

-          as i approaches infinity, the i positive integer sub-set
sequence value, for any positive integer value of n and k, respectively,
approaches infinity

4)      The infinite complete set of taxicab numbers, T(n,k)i, has
cardinality, aleph 0

https://simple.wikipedia.org/wiki/Aleph_null

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

Best regards, Larry.

On Sat, Jan 23, 2021 at 1:17 PM larry lacey <larryflacey at gmail.com> wrote:

> 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/
>>
>