# [seqfan] Something possibly OEISworthy. Wow.

David Wilson dwilson at gambitcomm.com
Tue Mar 10 15:34:10 CET 2009

```I actually found a sequence that may be OEISworthy, a table that has a
simple interpretation and ties together some loose sequences.

Let T(x,y) = the largest product of a partition of x into y positive
integers (1 <= y <= x).

It is easy to show that the distance between elements of P(x,y) is <= 1,
this determines P(x,y) to be:

P(x,y) = ([(x+k)/y] : 0 <= k < y)

giving the formula

T(x,y) = PROD(0 <= k < y; [(x+k)/y]).

Thus the table for T(x,y) starts

x    T(x, 1..x)
1    1
2    2,1
3    3,2,1
4    4,4,2,1
5    5,6,4,2,1
6    6,9,8,4,2,1
7    7,12,12,8,4,2,1
8    8,16,18,16,8,4,2,1
9    9,20,27,24,16,8,4,2,1
10    10,25,36,36,32,16,8,4,2,1
...

The columns tie together some loose sequences:

T(x,1) = x = A000027(x)
T(x,2) = A002620(x-2)
T(x,3) = A006501(x)
T(x,4) = A008233(x)
T(x,5) = A008382(x)
T(x,6) = A008881(x)
T(x,7) = A009641(x)
T(x,8) = A009694(x)
T(x,9) = A009714(x)

Also, these are pretty straightforward:

T(x,x-d) = 2^d = A000079(d) (d <= x/2)
MAX(1 <= y <= x, T(x,y)) = A000792(x)

The row sums:

1,3,6,11,18,30,46,73,111,170,254,392,574,868,1294,1933,2834,4267,6228,9312,...

are not in the OEIS.

I'm guessing there are some interesting properties. For example, the odd
elements appear to be those of index k^2 or 2k^2. I'm guessing
lim(n->inf, f(n+1)/f(n)) = sqrt(2).

```