On divisibility properties of certain multinomial coefficients. II (Q1434332)

By studying the divisibility properties of certain multinomial coefficients, in [J. Number Theory 48, No. 1, 80--87 (1994; Zbl 0810.11001)] \textit{G. Myerson} has introduced the function \[ f(n)= n!/L(n)[n/2]![n/3]!\cdots![n/b_k]~\dots, \] where \((b_k)\) is the Sylvester sequence defined by \(b_1= 2\), \(b_{k+1}= \prod^k_{i=1} b_i+ 1\) \((k\geq 1)\) and \(L(n)\) is the l.c.m. of \(1,2,\dots,n\). This function was further investigated by G. Myerson and J. W. Sander, who formulated also the following conjecture: for all \(m\geq 1\) there exists \(n_0\) such that \(m\) divides \(f(n)\) for all \(n> n_0\). They proved the validity of this conjecture for \(m\leq 8\), with related asymptotic estimates. The paper under review contains a proof of the conjecture for \(m\leq 100\). Another related result contains a partial answer to a reformulation of the above conjecture in terms of the least positive residues modulo \(\prod^k_{i=1} b_i\).