[seqfan] Something possibly OEISworthy. Wow.
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)
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:
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).
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