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

[David Wilson]

Here is a case where a more complete thought process leads to something 
more interesting:

We start with the proposed

%N A?????? Numbers of the form n^3 - (largest square < n^3)

Now, this is the range of the sequence

%N A?????? a(n) = n^3 - (largest square < n^3)

This has the generalization

%N A?????? a(n) = n^j - (largest k^th power <= n^j)
%N A?????? a(n) = n^j - (largest k^th power < n^j)
%N A?????? a(n) = (largest k^th power >= n^j) - n^j
%N A?????? a(n) = (largest k^th power > n^j) - n^j

Which might be interesting to add to the OEIS for a few small j and k.

It turns out that at least some of these are already in the OEIS, 
specifically, A053186 is the first sequence with j = 1 and n = 2.

The point here is that:

When considering a new sequence, consider the simpler sequences involved 
in constructing it, and consider generalizations. They might lead to new 
and interesting sequences.

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,
> 55, 56, 60, 63, 67, 74, 76, 79, 81, 83, 100, 104, 107, 109, 116, 127,
> 135, 139, 146, 147, 148, 150, 152, 155, 170, 174, 180, 184, 186, 191,
> 193, 200, 207, 212, 215, 216, 233, 235, 242, 244, 249 (list; graph;
> listen)
>
> Maximilian
>
>
> On Mon, Jan 5, 2009 at 12:55 PM, zak seidov <zakseidov at yahoo.com> wrote:
>   
>> Dear seqfans,
>>
>> 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, 60, 63, 67, 74, 76, 79, 81, 83, 100, 104, 107, 109, 116, 135, 139, 146, 147, 148, 150, 152, 155, 170, 174, 180, 184, 186, 191, 193, 200, 207, 212, 215, 216, 233, 235, 242, 244, 249, 251, 270, 277, 292, 293, 299, 301, 307, 308, 343, 355, 362, 364, 366, 368, 371, 391, 405, 424, 431, 433, 440, 448, 455, 459, 464, 471, 476, 496, 499, 503, 506, 508, 511, 515, 516, 524, 535, 546, 580, 586, 587, 589, 593, 596, 615, 622, 631, 639, 648, 652, 663, 667, 674, 676, 680, 683, 702, 703, 704, 726, 728, 732, 735, 755, 760, 764, 766, 767, 769, 775, 776, 782, 802, 804, 828, 831, 832, 847, 856, 859, 866, 868, 875, 888, 895, 900, 908, 914, 944, 945, 954, 964, 971, 973, 975, 980, 984, 991, 996, 999.
>>
>> What about filling gaps in this list
>> (finding more terms < 1000)?
>> thx, zak
>>
>> For n <~10^8 there are 207 values of n for which
>> 0<[n^3 - (largest square < n^3)]<1000, see
>> http://zak08.livejournal.com/3376.html
>> n^3 - (largest square < n^3)
>>
>> Cf. A077116  n^3 - A065733 (n),
>> A065733  Largest square <= n^3.
>>
>>
>>
>>
>>
>>
>>
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>     
>
>
> _______________________________________________
>
> Seqfan Mailing list - http://list.seqfan.eu/
>
>
>