Recursive formulae for the multiplicative partition function (Q1283425)

The identity \[ np(n)=\sum^n_{k=1} \sigma(k)p (n-k) \] is well-known. Using elementary means, the authors prove an analogous identity involving partitions of multi-partite numbers. They use this identity to correct some erroneous results given long ago by MacMahon. A couple of errors appear on the first page; for example, \((1,0)+(1,0)+ (1,0)\) should be \((1,0)+ (1,0)+(0,1)\); \(f(12)p(2,1)\) should be \(f(12)=p(2,1)\).