[seqfan] Re: Numbers not sums of perfect powers

[Richard Mathar]

I do not understand

bj> Numbers which are not the sum of squares-minus-1
bj> 2,4,5,7,8,10,13,16

in http://list.seqfan.eu/pipermail/seqfan/2010-April/004332.html :

There is
i)
"Numbers which have at least one partition with all parts in A005563"
that is
"Numbers which are a sum of squares-minus-1"
starting
0, 3, 6, 8, 9, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22
with a complement
1,2,4,5,7,10,13,.. (probably this is finite with 13 the last term)
Examples: a(3)=6 = A005563(1)+A005563(1). a(10)= 16 =A005563(2)+A005563(2).

There is
ii)
Numbers which are a sum of DISTINCT squares-minus-1
or rephrased
Numbers of the form A005563(i)+A005563(j)+..+A005563(k) with {i,j,...,k}
any set of distinct indices >=0:
0,3,8,11,15,18,23,24,26,27,32,35,38,39,42,43,46,47,48,50,51,53,56,58,59,..
Examples: a(6)=18 = A005563(1)+A005563(3). a(15)= 42 =A005563(1)+A005563(3)+A005563(4).

which has a complement
1,2,4,5,6,7,9,10,12,13,14,16,17,19,20,21,...

None of these two complements equals the proposed sequence.

RJM