A new function associated with the prime factors of (ⁿ_{𝑘})

Abstract

Let g ( k ) g(k) denote the least integer > k + 1 > k + 1 so that all the prime factors of ( g ( k ) k ) \left ( {\begin {array}{*{20}{c}} {g(k)} \\ k \\ \end {array} } \right ) are greater than k . The irregular behavior of g ( k ) g(k) is studied, obtaining the following bounds: k 1 + c > g ( k ) > exp ( k ( 1 + o ( 1 ) ) ) . {k^{1 + c}} > g(k) > \exp \,(k(1 + o(1))). Numerical values obtained for g ( k ) g(k) with k ≦ 52 k \leqq 52 are listed.