[seqfan] same sequence?
Jonathan Stauduhar
jstdhr at gmail.com
Fri Apr 19 20:08:48 CEST 2013
Hi all,
I proposing a family of sequences that count pairwise partitions of n in
various ways. Included in this family is A224708 (The number of
unordered partitions {a,b} of n, such that a and b are composite.) The
sequence begins: 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 1, 3, 1, 4,
2, 4, 2, 4, 2, 6,... (n=1,2,3,...)
Indexing just the odd numbers produces the subsequenceA224710 (The
number of unordered partitions {a,b} of the odd numbers 2n-1, such that
a and b are composite) which begins with 0, 0, 0, 0, 0, 0, 1, 1, 1, 2,
2, 2, 3, 3, 4,...
Alois P. Heinz notes that with n={5,7,9,..}, A224710 produces the sequence0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4,..., which forms A210469 (the number of odd composite parts of the pairwise partitions of 2n.) Is it obvious to someone why this would be true? Also, I don't see how A210469 can be used to produce a(1)=0 and a(2)=0 of A224710.
Thanks,
jonathan
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