Permutations, Fractions, Primes

[Leroy Quet]

Joseph Biberstine wrote:
>Naive Mathematica one-liner:
>Table[Max[Map[Length[Select[#, PrimeQ[#] &]] &, Map[Join[Numerator /@ #, 
>Denominator /@ #] &, Map[Table[Sum[1/#[[k]], {k, 1, m}], {m, 1, 
>Length[#]}] &, Permutations[Range[n]]]]]], {n, 1, 9}]
>
>Sequences begins (from n=1):
>{0, 3, 4, 4, 6, 8, 8, 10, 11}
>
>OEIS absent already.
>
>A naive maximum: 2n :)
>
>-JRB
>
>Leroy Quet wrote (in part): 
>> First, take a permutation {b(k)} of the first n positive integers, and 
>> then take the partial sums,...

>....
 
>> If we make a game of this, a better player might play the permutation
>> (2,6,1,4,5,3),
>> 
>> which gives the s(m)'s
>> 1/2, 2/3, 5/3, 23/12, 127/60, 49/20.
>> 
>> 
>> So we have the primes: 2,2,3,5,3,23,127, for a score of 7.
>> 
>> My question is, what is the maximum possible score for each given n?
>> 
>> I get the sequence beginning
>> 0,3,4,4,...

I see that Joseph Biberstine has computed an 8 for the 6th term.
So I was encouraged to find an n=6 permutation that gives a score of 8.
(Luckily, I found a permutation relatively quickly.)

2,6,1,3,5,4

The s(m)'s:
1/2,2/3,5/3,2/1,11/5,49/20.

(Yay!)

I would ask someone to compute more terms so this sequence can be 
submitted to the EIS, but, as is the case wih many game-derived 
sequences, the relatively long explanation of the sequence may make it 
too esoteric to be interesting to a general audience.

Thanks,
Leroy Quet