Mobius aMUsements

[Marc LeBrun]

Recently NJAS somehow inspired me to play at the ultra-simple minded game 
of negating all the primes in various constructions.

For example if you negate the primes in the prime factorization of n you get

   A61019:  1,-2,-3,4,-5,6,-7,-8,9,10,-11,-12,-13,14,15,16,-17,-18,-19,...

which turns out to be the inverse Mobius transform of

   A1615:    1,3,4,6,6,12,8,12,12,18,12,24,14,24,24,24,18,36,20,36,32,36,...

This sequence is described in the EIS as "Sublattices of index n in generic 
2-dimensional lattice; also index of GAMMA_0(n) in SL(2,z)."  (Anyone care 
to roughly decipher this definition?)

Similarly you can negate the primes in the classic sigma sum-of-divisors 
function, and get this new sequence of signed sums

   A61020:  1,1,2,3,4,2,6,5,7,4,10,6,12,6,8,11,16,7,18,12,12,10,22,10,...

and so forth.


This led to noticing that there is another sequence transform/inverse pair 
much like the Mobius transform, whose weights are "factors" of Mobius mu:

Transform:  Weight function:

Mobius      -1 to sum of exponents in prime factorization, except 0 when n 
is squarefull
MobiusInv   always +1, no exceptions

Maybeus     -1 to sum of exponents in prime factorization, no exceptions
MaybeusInv  always +1, except 0 when n is squarefull

For example MaybeusInv of 1,2,3,... gives us A1615 again, and applied to 
1,1,1,... seems to be

   A3444:  1,4,12,43,143,504,1768,6310,22610,81752,297160,1086601...

inscrutably described just as "Dissections of a polygon." (How so?)


Questions:

Are these transforms morally independent of the regular Mobius pair, or are 
they trivially related in some way?  (NJAS called mu "canonical"; can we 
really sing from a different hymnal?)

Generatingfunctionologically speaking, the Mobius transforms connect power 
series with Lambert series.  What sort of expansions are connected by the 
new pair?

Do these factored-mu weight functions appear anywhere?  If so, by what 
names?  (I've been calling them nu and psi).


If there's anything to this I'm thinking it might be fun to apply these 
transforms to the whole EIS corpus and see what crosslinks.  Some of the 
"hits" above encourage this, but the multiple appearance of A1615 might be 
a hint that maybe this isn't really worthwhile, because I'm overlooking 
some essential triviality.  So any additional insight you can provide would 
be most helpful...

Thanks!