Do points on elliptic curves exist where the denominators of point multiples grows more slowly than normal?

Question

Looking at prime multiples of $P=[1,1]$ on the curve $y^2=x^3+x-1$ the size of the denominator grows quite rapidly. So $5P=[\frac{685}{11^2},\frac{-18157}{11^3}],7P=[\frac{[154513}{443^2},\frac{-45623219}{443^3}]$

Looking purely at the x coordinate we have

k       sqrt(denominator(k*P))
3       1      
5       11      
7       443     
11      656243  
13      1048516201 
17      4451192987631289   
19      429330677071423079   
23      77700648618312971843469771251 
29      7517777790273327975717181745948599636777678033 
31      30820930821843565941522654527823658160891296876267921  
37      587308602722146793206557071567309610618121704728654635117962467133627539771     
41      22916712844996907328126003624983339852284728578820686400041407536873657173496053280584200979

I know this involves the Neron-Tate height somehow and is part of the reason finding a factor-base for calculating discrete logs on elliptic curves over $F_p$ seems impossible.

I was wondering if there is a choice of point or curve or prime multiple for which the size of the denominator of large prime multiples of that point would be small, or at least a partial factorization could be achieved.

Is this possible?

It would be very interesting to find a large composite number $n$ which could be shown to be a factor of the denominator of a a roughly $n^\frac13$-sized prime multiple of a point.