On a formula of Morita's partition function q(n) (Q1081627)

A formula is obtained for Morita's partition function q(n), where q(n)-1 is the number of conjugacy classes of \({\mathfrak sl}(2,{\mathbb{C}})\) in the Kac- Moody Lie algebra of type \(A^{(1)}_{n-1}\). Let \(n\in {\mathbb{Z}}_+\). A partition of n is a sequence \(\lambda =(\lambda_ 1,\lambda_ 2,...,\lambda_ r)\), where \(\lambda_ 1\geq \lambda_ 2\geq...\geq \lambda_ r\), \(\lambda_ i\in {\mathbb{Z}}_+\). Let \(a(\lambda)=\gcd (\lambda_ 1,\lambda_ 2,...,\lambda_ r)\) and \(f*g=\sum_{d | n}f(d) g(n/d)\). Morita's partition function is defined by \(q(n)=\sum a(\lambda)\), where the summation is over all partitions \(\lambda\) of n. It is proved that \(q(n)=\phi *p(n)\), where \(\phi\) is Euler's function and p(n) denotes the number of partitions of n.