[seqfan] Re: n^3 - (largest square < n^3)

[Benoît Jubin]

On Wed, Jan 7, 2009 at 4:27 PM, Maximilian Hasler
<maximilian.hasler at gmail.com> wrote:
> Dear SeqFans,
>
> I wondered why
>
> A106265 Numbers a such that equation Diophantine a+b^2=c^3 has integer
> solution(s) b and c.
>        2, 4, 7, 11, 13, 15, 18, 19, 20, 23, 25, 26, 28, 35, 39, 40, 44, 45,
> 47, 48, 49, 53, 54, 55, 56, 60, 61, 63, 67, 71, 72, 74, 76, 79, 81,
> 83, 87, 89, 95, 100, 104, 106, 107, 109, 112, 116, 118, 121, 124, 126,
> 127, 128, 135, 139, 143, 146, 147, 148, 150, 151, 152, 153
>
> is not the complement of
>
> A081121 Numbers n such that Mordell's equation y^2 = x^3 - n has no
> integral solutions.
>        3, 5, 6, 9, 10, 12, 14, 16, 17, 21, 22, 24, 29, 30, 31, 32, 33, 34,
> 36, 37, 38, 41, 42, 43, 46, 50, 51, 52, 57, 58, 59, 62, 65, 66, 68,
> 69, 70, 73, 75, 77, 78, 80, 82, 84, 85, 86, 88, 90, 91, 92, 93, 94,
> 96, 97, 98, 99
>
> Then I noticed that the cubes are missing in the former. I think they
> should be included, or the definition amended to disallow b=0.
> (Obviously c <= 0 is irrelevant, so one could say "...no solution in
> positive integers b,c".)

The definitions should be "Positive integers n such that..."?

This reminds me of a question I had: how to enter in the OEIS
sequences indexed by the integers?  For instance, for the identity, I
would write: 0,1,-1,2,-2... but apparently A001057 is not seen as
such.  There is also the trick consisting in using two entries, one
for a(n) and one for a(-n), but it's not very satisfying... Is it
worth creating a new keyword for these sequences (which are less
numerous)?

Here are useful tables about Mordell's equations:
http://www.lrz-muenchen.de/~hr/numb/mordell.html

Thanks,
Benoit



>
> Now a follow-up to the original post: (Apologies to those who consider
> the above as thread hijacking...)
>
> I thought of the "provers" sequence for
> A087285 Possible differences between a cube and the next smaller square.
>
> i.e. the (smallest) numbers b(n) such that b(n)^3 - next smaller
> square = A087285[n].
>
> First I had some trouble to find b(6) corresponding to A087285[6]=15,
> the number that was missing in Zak's list. This was the case when I
> implemented "next smaller" as "less or equal", viz
> n^3-[sqrt(n^3)]^2 =?= A087285[i].
> (This (l.h.s.) is actually the existing sequence A077116.)
> Then I realized that Zak's description explicitely uses " < " ; upon
> changing the code to
>
> vector(#A087285,i, { n=0; until(n^3-sqrtint(n^3-1)^2==A087285[i], n++); n } )
>
> the result is immediate, I just submitted it as
>
> %N A154332 Least positive integer m such that A087285(n) = A154333(m)
> = m^3 - next smaller square.
> %S A154332 3,2,32,15,17,4,7,6,35,8,11,10,14,21,12,28,65,9,56,18,136,568,23,99,101,
> (...)
>
> Regards,
> Maximilian
>
> PS: I would have been in favour of adding the initial term 1 to
> A087285 (but now my above submission relies on A087285[1]=2).
>
> On Mon, Jan 5, 2009 at 3:11 PM, Maximilian Hasler wrote :
>> See
>> A087285 Possible differences between a cube and the next smaller square.
>>        2, 4, 7, 11, 13, 15, 19, 20, 26, 28, 35, 39, 40, 45, 47, 48, 49, 53,
> (...)
>> On Mon, Jan 5, 2009 at 12:55 PM, zak seidov <zakseidov at yahoo.com> wrote:
>>> There are 158 possible non-zero values of difference
>>> [n^3 - (largest square < n^3)] less than 1000:
>>>
>>> 2, 4, 7, 11, 13, 19, 20, 26, 28, 35, 39, 40, 45, 47, 48, 49, 53, 55, 56,
> (...)
>
>
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