Re: Convergence of reciprocal odd perfect numbers
- --- In email@example.com,
"djbroadhurst" <d.broadhurst@...> wrote:
> The asymptotic number of perfect numbers less than x was proven inThe really tough question is whether the sum of the
> MR0090600 (19,837d)
> Hornfeck, Bernhard; Wirsing, Eduard
> Über die Häufigkeit vollkommener Zahlen. (German)
> Math. Ann. 133 (1957), 431-438.
> to be less than x^eps for all eps>0.
> From this it trivially follows that the sum of the reciprocals
> of the perfect numbers is finite.
> I regard Carl's result in
> MR0618552 (82m:10012)
> Pomerance, Carl
> On the distribution of amicable numbers. II.
> J. Reine Angew. Math. 325 (1981), 183-188.
> as "stronger" since it proves the finiteness of
> of the sum of the reciprocals of the amicable numbers.
reciprocals of the sociable numbers is finite.
One might guess so, from the opinions in
but there seems little hope of a proof,
given that we do not know whether an integer as small
as 276 is sociable.