[seqfan] Re: Upper bound for A091895 and A091896
Michel Marcus
michel.marcus183 at gmail.com
Tue Apr 5 16:51:32 CEST 2022
Thank you both.
A bit different, did you see A351913 ?
Any idea when can we say that Unknown can be set to -1 ?
MM
Le mar. 5 avr. 2022 à 06:31, M. F. Hasler <oeis at hasler.fr> a écrit :
> 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
>
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