[seqfan] Average ratio of terms of a sequence

[john.mason at lispa.it]

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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