[seqfan] Does A330737 fly or will it crash? And is A199337 well-defined?

[Antti Karttunen]

Cheers,

prompted by sequence A199337, "Number of highly composite numbers not
divisible by n":
https://oeis.org/A199337
I started wondering how to actually compute that.

So I created a tentative auxiliary (or "probing") sequence A330737
whose definition is:
   a(n) is the position of first index k in A002182 (highly composite
numbers) from which onward all terms A002182(i), i >= k, are multiples
of the n-th prime, a(0) = 1 by convention.

whose draft is here: https://oeis.org/draft/A330737

Is that sequence well-defined?
That is, is it guaranteed that for any n and big enough k, prime(n) |
A002182(i), for all i >= k ?

And what about A199337?
That is, it is guaranteed that for any n and big enough k, n |
A002182(i), for all i >= k ?

Moreover, I wonder if there is a theorem that
For any n and big enough k, A002182(n) | A002182(i), for all i >= k ?

But in practice, how we can compute both sequences A199337 and A330737
with some certainty and for how far?


Best regards,

Antti