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

Maximilian Hasler maximilian.hasler at gmail.com
Wed Jan 7 16:27:51 CET 2009

```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".)

Now a follow-up to the original post: (Apologies to those who consider

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,
(...)

```