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

[M. F. Hasler]

On Thu, Jan 2, 2020 at 2:10 PM Antti Karttunen wrote:

> prompted by sequence oeis.org/A199337, "Number of highly composite
> numbers not divisible by n",
> I created oeis.org/draft/A330737: 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.



is it guaranteed that for any n and big enough k, prime(n) | A002182(i),
> for all i >= k ?



it is guaranteed that for any n and big enough k, n | A002182(i), for all i
> >= k ?
>
> For any n and big enough k, A002182(n) | A002182(i), for all i >= k ?
>

The 1st and 3rd are a consequence of the 2nd: just take n to be prime(n)
resp A2182(n).
To prove the 2nd we show that for any prime p and any exponent e_p,
eventually p^e_p will divide all sufficiently large N = A2182(n).

This follows from the formula
[ log P / log p ] <= e_p < = 2 [ log P' / log p ]
(eq.(54) in Ramanujan (1915), doi:10.1112/plms/s2_14.1.347
<https://doi.org/10.1112/plms/s2_14.1.347>),
with
p = any prime in the factorization of N,
e_p = the corresponding exponent,
P = largest prime factor of N, P' = next larger prime.
[.] is as usual the floor function.

Indeed, first this gives an upper bound for each exponent e_p as a function
of the largest prime factor P. As a consequence, P cannot be bounded as N
-> oo.

Then, it gives a lower bound for each e_p, as function of P, so that e_p ->
oo for each p, QED (if I'm not wrong).

(This also answers a question I asked some time ago in A002182.)

- Maximilian