A000028

[franktaw at netscape.net]

Sequence A000028 is interesting.
 
One question - this sequence has links to A000201 and A001950 (Wythoff sequences).  Why?  There is no obvious connection.  If there is a real connection, a comment should be added to A000028 describing it.
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I also note that the presence of a number in A000028 depends only on its prime signature.  If I haven't made any mistakes, the signatures present through products of 8 primes are:
[1]
[2]
[1^3]
[4] [3,1]
[2^2,1] [1^5]
[4,1^2] [3,1^3]
[7] [6,1] [5,2] [4,3] [3^2,1] [2^2,1^3] [1^7]
[4,2^2] [4,1^4] [3,1^5] [2^4]
 
There is no obvious pattern here.  The sequence of the number of partitions of n present is:
1,1,1,2,2,2,7,4,...
which is not in the OEIS.  (I'd want a few more terms before I submit it, though.)  The complementary sequence, for A000379, is:
1,2,3,5,9,8,18,...
which is also not in the OEIS.
 
Incidently, how would one add the set of partitions above itself to the OEIS?  Just concatenating them, there is no way to tell where one ends and the next begins.  Two ideas: (1) terminate each partition with a zero:
1,0,2,0,1,1,1,0,4,0,3,1,0,2,2,1,0,1,1,1,1,1,0,...
or (2) submit two sequences, with the partitions concatenated and their lengths:
1,2,1,1,1,4,3,1,2,2,1,1,1,1,1,1,...
1,1,3,1,2,3,5,...
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I would also be inclined to add a reference from A000028 to A033627 (with an appropriate comment); this is the exponents of the prime powers in A000028 (i.e., the singleton partitions in the list above).
 
Franklin T. Adams-Watters
16 W. Michigan Ave.
Palatine, IL 60067
847-776-7645
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