A076031 alt.(1) (2 8) (3 4 12) (5 7 10 14) (6 9 15 18 20.)

[Don McDonald]

In message <DCA89D93-F434-11D6-B3AD-003065438114 at rogers.com> you write:

> > %I A076031
> > %S A076031 1,2,8,3,4,12,5,6,7,210,9,10,11,12,330,13,14,15,16,17,46410
> > %N A076031 Group the natural numbers so that the n-th group contains 
> > the smallest set of n numbers whose product is a s\
> >   quare: (1), (2, 8), (3, 4, 12), (5, 6, 7, 210), (9, 10, 11, 12, 
> > 330), (13, 14, 15, 16, 17, 46410), ...
> > %e A076031 1; 2,8; 3,4,12; 5,6,7,210; 9,10,11,12,330; 
> > 13,14,15,16,17,46410; ...
> > %Y A076031 Cf. A076027, A076028, A076029, A076030.
> > %K A076031 more,nonn,tabl
> > %O A076031 1,2
> > %A A076031 Amarnath Murthy (amarnath_murthy at yahoo.com), Oct 01 2002
> 
> I'm not comfortable with the lack of specificity of the word 
> "smallest". Are we constrained merely to using (in the nth grouping) 
> the smallest still-available (n-1) terms? [Other, more interesting, 
> constraints come to mind. Some other time, perhaps.]
>
Are the following an improvement? Especially the first.

(1) (2 8) (3 4 12) (5 7 10 14) (6 9 15 18 20.)  compA76031
1^2,  4^2,   12^2,   70^2,       540^2, 
                                                
(1) (2 8) (3 7 21) (4 5 6 30) (9 10 12 15 18.)  compB76031
1^2,  4^2,  ** 21^2,   60^2,       540^2,
(missed 12^2, so we need
a rule or algorithm that favours the previous line.
i.e. the following.)

Factorise suitable squares as far as possible such that
no integer occurs more than once and the nth suitable square
is a product of n natural numbers.

Don McDonald.
11.11.02  10:01
my file >  McD.Calc.Profile.eisintegsq.Seqfan.groupsquar