Put s ( 1 ) ( n ) = σ ( n ) − n , σ ( n ) = Σ d / n d {s^{(1)}}(n) = \sigma (n) - n,\sigma (n) = {\Sigma _{d/n}}d . s k ( n ) = s ( 1 ) ( s ( k − 1 ) ( n ) ) {s^k}(n) = {s^{(1)}}({s^{(k - 1)}}(n)) . In this note we prove that for every k the density of integers satisfying \[ s k ( n ) = ( 1 + σ ( 1 ) ) n ( ( σ ( n ) − n ) / n ) k {s^k}(n) = (1 + \sigma (1))n{((\sigma (n) - n)/n)^k} \] is 1. Several unsolved problems are stated.