[seqfan] Re: Upper bound for A091895 and A091896

[M. F. Hasler]

On Sun, Apr 3, 2022 at 12:05 PM Hugo Pfoertner wrote:

> A bound of k <= 2*n^2 should be sufficient. This covers the extreme cases
> n=2 with d(8)/8 = 4/8 = 1/2 and n=6 with d(72)/72 = 12/72 = 1/6. The factor
> f=2 in the required bound for k to exclude d(k)/k=n decreases for larger n.
> E.g, f=5/3 at n=12, f=6/5 at n=20, f=20/21 at n=42, f=24/35 at n=70, f=8/15
> at n=90, f=24/55 at n=110, ....


I agree. Actually, not only the factor 2 might be decreased,
but a smaller power of n might be sufficient :
Up to n = 10^5,  a(n)/n <= 16 n^(1/3).
The record of the ratio a(n)/n   for n <= 10^5  is  672  at  n = 90090.
I suggest considering the sequence  a(n) / n = A352834(n)  [proposed draft].

- Maximilian