A tree sequence

[koh]

    I wrote :  

>    Seqfans
>
>    I considered about a sequence related with tree graph.
>
>    [Figure]
>
>         |                  m=1
>
>       __|__                m=2
>         |
>
> __|_____|_____|__          m=3
>   |     |     |
>         |
>       __|__
>         |
>  
>   _|_    x    _|_         m=4
>_|__|_____|_____|__|_
> | _|_    |    _|_ |
>    |     |     |
>      _|__|__|_
>       | _|_ |
>          |
>          y

>    How do they say these graphs?
>    I don't know the name which mathematicians ordinarily use.
>    
>    I suppose that "Tree graph of degree 3 of length n".
>    Is it correct?         
>
>    %I A000001
>    %S A000001 1,3,81,1594323,12157665459056928801
>
>    %N A000001 Number of partitions into "Bus route" of G_n.
>    %F A000001 a(n)=3^((3^n-1)/2) n={length of tree graph}-1
>    %e A000001 Figure of G_3.
>
>   _|_         _|_         
>_|__|_____|_____|__|_
> | _|_    |    _|_ |
>    |     |     |
>      _|__|__|_
>       | _|_ |
>          |
>
>    %Y A000001 A131709
>    %K A000001 none
>    %O A000001 0,2
>    %A A000001 Yasutoshi Kohmoto   zbi74583 at boat.zero.ad.jp
>
>
>
>
>    Yasutoshi
>    

    Neil wrote : 

>Dear Yasutoshi,  Before I add that sequence to the OEIS,
>could you please tell me how G_n is defined?  The rule
>is not clear to me!  G_1 and G_2 have 4-fold symmetry,
>but G_3 does not.

    The rule is easy.
    Suppose that you walk from the point x to y on G_3. See Figure m=4.
    You will pass three cross-point which have three branches always.
    If you go to the other terminals then the numbers of branches are the same.
    I mistook the degree of G_n and the length.
    It is 4 degree and the length is n+1 or m.

    I mean the tree graph G_n as follows.

    A trunk has 3 branches.
    Each branches have 3 sub-branches.
    Each sub-branches have 3 sub-sub-branches.
    .....
    The length from the terminal of the trunk to the terminal of sub-.....sub branch is n+1.    

    The number of nodes whose degree are 4 in G_n is 1+3+3^2+3^3.....
    So it becomes (3^n-1)/2.
    And number of partitions into bus route of G_n is {matching number of 4}^{number of nodes whose degree are 4}.

    A tree like this graph really exists.
    "Mitsumata" which means three branches is a material of "Washi" which is a kind of paper.
    A photo is here.
    http://image.myany.jp/org/6c511f3698cf6db33d63d6450f331344.jpg
    
    You recognize three branches.

    Yasutoshi   
    




A134592 defines a sequence where n->a(n) assigns "the largest highly
composite number not a multiple of n" to the indices n. I do not understand
how this works..
Image the standard prime decomposition of any number,
2^e_1*3^e_2*5^e_3*7^e_4*..
where base factors are the primes in ascending order and exponents positive
numbers (including 0). Look at this as a bar chart with primes along the
horizontal axis and a mountain range above with hills and valleys as high as
the e_i.
These ranges have finite width, extending to the right according to
the largest prime factor of the number, where all e_i to the right are zero
(an infinite horizontal base plane right to the mountain range).

Highly composite numbers (HCN's) A002182 have mountain ranges which are flat or
descending as one travels over the mountains from the left (prime 2) to the
right. (This is a "single water shed" requirement, so water could flow from
the left to the right downstream without being trapped by a valley.)

The requirement on a(n) not being a multiple of n means that "n" must have
at least one exponent of the prime decomposition  being larger than the
exponent of the HCN at the same place; so the mountain range of "n" must
peak above the mountain range of "HCN" at least once.
Supposed one such associated HCN has been found and supposed the
rightmost, nonzero e_i of "n" is larger than 1 (examples n=2*3^2, n=7*11*13^5),
then I can always find larger HCN's than any one found by extending the
mountain range of the HCN to the right by multiplication of any number of
primes that are larger than that largest prime of the prime decomposition
of "n".

candidate which falls below the mountain range of the stars at least at
one place: I can go on with "pluses" forever on the right, increasing
the HCN candidate ad libidum...

e  |

Simplest example at n=4=2^2: take the HCN's 2,2*3, 2*3*5,2*3*5*7 etc,
with no "largest" in sight. (The current value in the OEIS is 2*3=6.)
So the question is: how well defined is that sequence?

Richard



Richard Mathar <mathar at strw.leidenuniv.nl> wrote:
:
:A134592 defines a sequence where n->a(n) assigns "the largest highly
:composite number not a multiple of n" to the indices n. I do not understand
:how this works..

I think the intended sense of the definition is "largest highly composite
number (definition 1) not a multiple of n", ie, largest element of A2182()
not a multiple of n.

If I correctly understand it, for all k there exists an n such that
A2182(n+i) is divisible by k for all i >= 0, so by this definition
A134592(n) would be well-defined.

Assuming this is correct, I would suggest adding "(definition 1)" to the
name and adding a comment clarifying the reference to A2182():

%I A134592
%S A134592 1,4,6,48,4,720,1260,1680,48,50400,6,665280,720,48
%N A134592 a(n) = largest highly composite number (definition 1) not a multiple of n.
%C A134592 I.e. largest element of A002182() not a multiple of n.
%Y A134592 Cf. A002182, A022404.
%K A134592 more,nonn,new
%O A134592 2,2
%A A134592 jhbubby(AT)mindspring.com, Jan 24 2008

Hugo