[seqfan] Thanks and another question

David Newman davidsnewman at gmail.com
Sun Jun 29 04:24:41 CEST 2014

Thanks to those who took the time to answer my request for help.

This sequence led me to another sequence, A060692, which is "the number of
parts if 3^n is partitioned into parts of size 2^n as far as possible and
into parts of size 1^n.

Two formulas are given:

a(n) = floor(3^n/2^n) + (3^n mod 2^n)


a(n)= 3^n (mod 2^n-1)  [Alex Ratushnyak Jul 22 2012]
This second formula would seem to imply that a(n)< 2^n.

Now on to the Wikipedia article for Waring's Probllem.

The article claims that: "No values of k are known for which 2^k { (3/2)^k}
+ [(3/2)^k]>2^k ."

Here the curly brackets indicate the fractional part.

Large searches are mentioned, but it would seem that no proof is known.
Yet the OEIS comment by Ratushnyak would seem to imply that the answer is
known. The quantity is never greater than 2^k because it is less than 2^k.
 Where am I going wrong?

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