[seqfan] Re: Average ratio of terms of a sequence

[john.mason at lispa.it]

2 corrections.
1. The question marks in the last sentence should have been double-quotes
2. The values indicated in the last sentence are not the ratios between
adjacent terms themselves but the limits toward which these ratios
apparently converge
John

Inviato da iPad

> Il giorno 21 Jun 2016, alle ore 16:42, john.mason at lispa.it ha scritto:
>
> Dear Seqfans
>
> (Please will someone let me know if this is old hat.)
>
> Define the avgratio(n) for a specific sequence as being the mean of a
(i)/i
> for i=1 thru n.
>
> Consider in particular those sequences that are permutations of the
> positive integers. So for such sequences, avgratio(n) >= 1 for all n.
>
> For A000027, the sequence of positive integers, avgratio(n) is always 1.
>
> For A254077, which is conjectured to be a permutation of the positive
> integers, avgratio(n) appears to converge to 1. avgratio(10^9) is about
> 1.0006.
>
> On the other hand, it is possible to define sequences with non-converging

> avgratio. Consider the example S such that: S(n), for  non-prime n, is
> 10^n; S(n), for prime n, is the first positive integer not already in
> sequence.
> So S starts with terms:
> 10,1,2,10000,3,1000000,4,100000000,1000000000,10000000000.
> S is a permutation of the positive integers, but with diverging avgratio.
>
> It is also possible to define sequence T_x, a permutation of the positive

> integers, for any real x >= 1, such that avgratio(n) will converge to x.
> Define T_x thus: T_x(1)=1; for n > 1, calculate r = avgratio(n-1); then
if
> r>=x, T_x(n)=first positive integer not already in sequence; if r<x, then

> T_x(n)=  first positive integer not already in sequence such that
> avgratio(n) >= x
> Thus T_2 starts with terms:
> 1,6,2,14,3,20,4,27,5,34,7,41,8,47,9,54,10,62,11,68 (not currently in
> OEIS).
> As can be seen, the terms alternate between ?low? and ?high? and have
> ratios between successive terms: 3+2*sqrt(2) and 3-2*sqrt(2), which have
> sum = 6 and product = 1.
>
> john
> __________________________________
>
>
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