Explicit Infinite Families of Palfree Squares, Cubes, and Fourth Powers

A positive integer is called palfree here if its ordinary decimal expansion contains no contiguous palindromic factor of length at least two. We give explicit unbounded families of palfree squares and cubes and derive consequences for OEIS A052061--A052068. For squares, the numbers\[x_t=\frac{4(10^{9t}-1)}{9},\qquad t\ge 1,\]have palfree squares. For cubes, we prove the stronger all-index statement\[\left(\frac{10^n-1}{3}\right)^3\]is palfree for every integer \(n\ge 1\). The cube proof uses exact three-digit block decompositions according to \(n\bmod 3\). These constructions prove that the square- and cube-root sequences A052061 and A052063 and the corresponding value sequences A052062 and A052064 are infinite. Because the two root families are unbounded, the least qualifying root above \(10^n\) exists for every \(n\ge 1\) in both the square and cube cases. Consequently, A052065, A052066, A052067, and A052068 are defined at every index and are infinite. We also give an explicit infinite family of palfree fourth powers,\[\left(\frac{2(10^{9t}-1)}{3}\right)^4,\qquad t\ge 1,\]using an exact nine-digit block decomposition. The infinite statements are proved algebraically; finite computation is used only as a consistency check.