[seqfan] Numbers not sums of perfect powers

[Benoît Jubin]

Dear SeqFans,

I plan to submit the following finite, full sequences, motivated by
the study of replicate tilings (see concurrent discussion, still in
need of two proofs).  Any comments are welcome (and there is a proof
request at the end).

Numbers which are not the sum of exactly one square and a a sum of
squares-minus-1
2,3,5,6,8,11,14
(motivated by certain replicate tilings, where each tile can be
replaced by a squared number of tiles)

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

Numbers which are not the sum of squares larger than one (already in
the OEIS: A078135)
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 19, 23

Next: why not generalize to any exponent ?  It would be nice to submit
the corresponding sequences for cubes and to prove finiteness for any
exponent, as comments.  I don't have a proof for the first two, and I
don't have a computer of my own currently so can't do any programming.

For analogues of the third sequence, we have:
The numbers which are not the sum of k^th powers larger than one are
exactly those in [1,6^k-3^k-2^k] but not of the form
2^k.a+3^k.b+5^k.c.  This relies on the following fact: if m and n are
relatively prime, then the largest number which is not a linear
combination of m and n with positive integer coefficients is mn-m-n.

Is there an analogous result for relatively prime integers {m_i} ?

Benoit