Permutations, Fractions, Primes

[Joseph Biberstine]

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:
> First, take a permutation {b(k)} of the first n positive integers, and 
> then take the partial sums, for 1<=m<=n,
> 
> s(m) = sum{k=1 to m} 1/b(k).
> 
> After reducing the n fractions {s(m)} so their numerators and 
> denominators are coprime,
> we can get a "score" equal to the total number of primes in both the 
> numerators and denominators of the s(m)'s.
> 
> For example, if, for n=6, we have the permutation
> (3,4,1,6,5,2),
> 
> then we have the s(m)'s
> 1/3, 7/12, 19/12, 7/4, 39/20, 49/20.
> 
> So we have the primes: 3,7,19,7, for a score of 4.
> 
> 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,...
> 
> Thanks,
> Leroy Quet