Primes in a sparse sequence (Q1208188)

The author considers the partial sums \(s_ n=\sum_{1\leq r\leq n}r^ r\) of the sequence \(n^ n\) that had been shown to be ultimately periodic mod \(m\), for each \(m\), by \textit{R. Hampel} [Ann. Polon. Math. 1, 360-366 (1955; Zbl 0065.028)]. Using some of Hampel's results the author is able to estimate how often \(s_ n\) is divisible by a suitable squarefree shifting modulus \(d\), and is thus able to apply Selberg's sieve to derive an upper bound \(\ll(\log x)/(\log\log x)^ 2\) for the number of prime \(s_ n\) not exceeding \(x\).