# [seqfan] Stirling-2 Numbers and dropping times of the Collatz Conjecture

Sun Jun 27 23:38:32 CEST 2010

Hi all!

I am contemplating whether the following strange connection
between the Stirling numbers of the second kind S(n,m) and
the Collatz conjecture holds. Let me state it as a conjecture:

/
| If the value of m maximizing m!S(n,m) equals
| M := floor(1 + (n+1)/log(4)) then M is a
| "dropping time" of the Collatz (3x+1) iteration.
\

See A019538 for m!S(n,m).
See A002869 for max in the n-th row of the above.
See T. D. Noe's A122437 for allowable values of
the "dropping time" of the Collatz (3x+1) iteration.

The sequence of the M's such that max = M, starts
2,7,12,20,25,38,51,56,64,69,82,95,100,108,113,121,
126,139,144,152,157,165,170,183,188,196,201,209,214,
227,232,240,245,253,258,271,276,284,289,297,

Arguments n of the above sequence
3,10,17,28,35,53,71,78,89,96,114,132,139,150,157,
168,175,193,200,211,218,229,236,254,261,272,279,
290,297,315,322,333,340,351,358,376,383,394,401,412,
Is this, for a(n)>10, a subsequence of A048265?
a subsequence of A159843?

Cheers, Peter

http://en.wikipedia.org/wiki/Collatz_conjecture