[seqfan] Re: Array of "higher order" Landau-function sequences.

Antti Karttunen antti.karttunen at gmail.com
Fri May 10 19:36:50 CEST 2013

On Fri, May 10, 2013 at 5:48 PM, Antti Karttunen
<antti.karttunen at gmail.com> wrote:
> Hello all,
>
> would this kind of array make any sense:
>
> As the first row we have Landau's function http://oeis.org/A000793
>
> as the second row a sequence giving the function for the maximum value
> of LCM(A000793(n),LCM_of_some_partition_of(n))
> (where "some_partition" is for n>3 some other partition than which
> maximized the value for the same n, i.e. that which Landau's function
> "used").
>
> as the third row a sequence giving the function for the maximum value
> of LCM(that_previous_second_row(n),LCM_of_some_partition_of(n)),
> etc.
>
> It should be clear that every column will eventually set to a constant
> value, after all the available partitions of n "have been exhausted".

Also, of interest should be the number of distinct values in each column n.
(i.e. the time to reach the fixed point with such a process, always
choosing yet another partition p1+p2+...+pk of n, such that
LCM(p1,p2,...,pk,the_corresponding_value_from_previous_row(n)) is the
maximum possible)
I think it should be less than http://oeis.org/A009490 (because we may
not use any "subset-partitions") and should depend on the prime
signature of n? (But not only on that?)

Another idea, a kind of "LCM-variant" of http://oeis.org/A212721

Array where each row (or column for transposed version) gives the
LCM's of partitions of n, ordered in such way that we always start
from n copies of 1 (yielding product 1), then choose the next
partition so, that the next LCM-product / this LCM-product is the
least possible (so the next partition is 2 + (n-2) copies of 1, giving
LCM-product 2, with the ratio 2), the next partition is 3 + (n-3)
copies of 1, giving LCM-product 3, and the ratio 3/2, and so on, up to
the singular partition {n}, only after which there will be anything
interesting.
E.g. for 8, we should have 1,2,3,4,5,6,7,8,10 (= LCM(1,2,5)),12 (=
LCM(1,3,4), 15 (= LCM(3,5))

A near match for that can be found at:
http://oeis.org/A222417
(but there is an extra 24 for the eight row).

Yours,

Same

>
> Sequence A225558 I just submitted (or some of its referents) might
> have something to do with this, or more probably not. As usual, I
> can't think clearly now.
>
>
> Yours cordially,
>
> Antti Karttunen