/
| def Moebius(n,k) : return moebius(n//k) if k<>0 and n%k == 0 else 0
\

Now Paul looked at the number of k such that the first forward
difference in k does not vanish: Moebius(n, k+1) <> Moebius(n,k).

def A123709(n) :
return len([k for k in (1..n-1) if Moebius(n,k+1) <> Moebius(n,k)])+1

Et voilà, three more of Paul's sequences are easily described:

def list(n,m)       : return [i for i in (1..n) if A123709(i) == m]
def A123710_list(n) : return list(n,4)
def A123711_list(n) : return list(n,8)
def A123712_list(n) : return list(n,16)

In particular A123712(n) now simply means
   card {0<k<n | Moebius(n, k+1) <> Moebius(n, k)} = 15.

On the other hand Reinhard found in A178212 a sequence
with the description: {n | Omega(n) > 3 and omega(n) = 3}
which is by numerical experiment equal to A123712.

So now we can formulate the A123712/A178212-conjecture:

/
| Let n>=1 be an integer. Then Omega(n) > 3 and omega(n) = 3
| if and only if
| card { 0<k<n | Moebius(n,k+1) <> Moebius(n,k)} = 15.
\

Of course this '15' looks rather strange here. On the other hand we
certainly see here only the tip of an iceberg and the strangeness
will go away as soon as the general relation is found.

Any idea how this general relation might look like?