[seqfan] Re: square unit fractions

[hv at crypt.org]

I have also posted the conjecture to stackexchange here:
  https://math.stackexchange.com/questions/4862230/

Hugo

I wrote:
:Apologies for the length of this; there are three parts pertaining to the
:same function: one on a conjecture, one on a sequence, and one on calculation.
:
:Consider a set S of positive integers, and define:
:  u(S) = sum_{s in S}{1/s}
:  p(S) = sum_{s in S}{1/p_s}  the s'th prime
:  q(S) = sum_{s in S}{1/s^2}
:
:Conjecture:
:
:It is straightforward to show by a greedy algorithm that for any rational r,
:there exists an S with u(S) = r. Conversely, for almost all r there does
:not exist an S with p(S) = r: at most one rational with any given
:denominator can be reached (and none unless the denominator is squarefree).
:
:Letting \rho = \pi^2/6, q(S) falls in the ranges [0, \rho-1), [1, \rho).
:I conjecture that for any rational r within those ranges there exists an S
:with q(S) = r. Can anyone prove or disprove this? (This would seem an
:interesting result, since squares are notably less dense than primes
:within the integers.)