A086833

[franktaw at netscape.net]

While that might be an interesting sequence, it seems unlikely that that is what the author intended for this sequence.  In particular, a(2^m) = m for the values shown (m>0), and one could hardly believe that that was the last addend of any shortest addition chain for 2^n.
 
I think I may have figured this out.  Look at all shortest addition chains for n, and express them in this sum form.  Count how many different addends appear in each chain, and take the minimum.  This at least matches the first few values and the 2^n case.
 
There is, of course, a problem with this definition: not every addition chain can be put into this form; it assumes that each term adds to the previous term (i.e., that we are dealing with a Brauer chain).  12509 is the smallest number which does not have a shortest addition chain with this property.
 
If this is the intended meaning, somebody should verify and extend the sequence, and also fix the description.
 
Franklin T. Adams-Watters
 
-----Original Message-----
From: David Wilson davidwwilson at comcast.net


I would guess Hugo is correct in his interpretation that the sequence was intended to give the smallest final addend among the shortest addition chains for n, which is a potentially interesting concept. I presume the author is not a native English speaker, which may account for the obscure description, and that the example and value for a(23) are errors, a condition not unknown in the OEIS. It might be enlightening to compute the sequence of actual smallest final addends among minimal addition chains for n and compare this sequence to the sequence given. 
 
----- Original Message ----- From: "Pfoertner, Hugo" <Hugo.Pfoertner at muc.mtu.de> 
 
> -----Original Message----- 
> From: franktaw at netscape.net [mailto:franktaw at netscape.net] 
> 
> 
> The description for A086833 is "Shortest addition chain's minimum member." 
> Does anyone know what this means? I can't make any sense out of it, and the 
> example doesn't really help any. 
> 
> Franklin T. Adams-Watters 
> -------------------------------- 
> 
> The sequence is a candidate for the "obsc" keyword. The example "a(23)=5 
> because 23=1+1+2+1+4+9+5" seems to indicate something like "difference 
> between final result and previous partial sum", but there are 4 different 
> addition chains for n=23. See A079300. 
> 
> The one in the example is 1 2 4 5 9 18 23 with 23-18=5, but another one is 1 
> 2 3 5 10 20 23. Where does a(23)=5 fit here? 
> 
> BTW, one example of an addition chain for each n<=2048 is given in 
> http://www.randomwalk.de/sequences/addchains.txt 
> 
> Hugo Pfoertner  
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