%N Numbers n such that 2n/sigma(n) - 1 = 1/x for some integer x.
%C If the number x is a prime which does not divide n, then n*x is a perfect number. This happens (so far) only when x = 2n-1 = 2^p-1 is a Mersenne prime (cf. A000043). But if x does not divide n, as, e.g., for (n,x)=(10,9), then n*x is a so-called freestyle perfect number, cf. A058007: Namely it "would be perfect if x is assumed to be prime", which means that sigma(n*x) is replaced by sigma(n)*(x+1) in the relation 2P=sigma(P) characterizing perfect numbers P, listed in A000396.
%C See also the (more interesting) subsequence of odd terms, A222263.
%H Donovan Johnson, <a href="/A222264/b222264.txt">Table of n, a(n) for n = 1..1000</a>
%e 8 is in the sequence because 2 * 8/sigma(8) - 1 = 16/15 - 1 = 1/15.
%e 9 is not in the sequence because 2 * 9/sigma(9) - 1 = 5/13.
%e 10 is in the sequence because 2 * 10/sigma(10) - 1 = 20/18 - 1 = 1/9.
%t Select[Range[10^5], IntegerQ[2#/DivisorSigma[1, #] - 1] &] (* _Alonso del Arte_, Feb 20 2013 *)
%o (PARI) for(n=1,9e9, numerator(2*n/sigma(n)-1)==1 & print1(n","))
%A _M. F. Hasler_, Feb 20 2013