On the equation \(\tau (\lambda (n))=\omega (n)+k \) (Q2470257)

From the text: For every positive integer \(n\), the function \(\tau(n)\) counts the number of divisors of \(n\), the function \(\omega(n)\) counts the number of distinct prime divisors of \(n\), while the Carmichael function \(\lambda(n)\) is the exponent of the multiplicative group of the invertible congruence classes modulo \(n\). We investigate some properties of the positive integers \(n\) that satisfy the equation \(\tau(\lambda(n))=\omega(n)+k\) providing a complete description for the solutions when \(k = 0, 1, 2\), and giving some properties of the solutions in the other cases. For a set \(\mathcal A\) of positive integers write \(\mathcal A(x) = \mathcal A \cap [1, x]\) and let \(p\) and \(q\) be prime numbers. Let us set \[ \mathcal A_k = \{n: \tau(\lambda(n)) = \omega(n) + k\}. \] We show that if \(k\) is a positive integer and \(b_k = 2(k + 1)^2 + 3 + \lfloor\log_2(2(k + 1)^2 + k + 1)\rfloor\), then the upper bound \[ \#\mathcal A_k(x)\ll_k \frac{x(\log \log x)^{b_k}}{(\log x)^2} \] holds as \(x\to\infty\). Furthermore, we show that if \(k > 4\), then the lower bound \[ \#\mathcal A_k(x)\gg_k\frac x{(\log x)^2} \] holds as \(x\to\infty\). We also give a complete description on the sets \(\mathcal A_0,\mathcal A_1\) and \(\mathcal A_2\). We show that \(\mathcal A_0\) contains 8 integers, \(\mathcal A_0 = \{2, 6, 12, 24, 30, 60, 120, 240\}\), while the infiniteness of \(\mathcal A_1\) and \(\mathcal A_2\) would follow if it were known that there exist infinitely many primes of the form \(2q +1\) with \(q\) also prime. Finally, we deal with the cases \(k=3,4\) proving that if either \(\mathcal A_3\) or \(\mathcal A_4\) are infinite then there exists an even positive integer \(c\) such that the set of primes of the form \(p=cq^\beta+1\), with \(q\) prime and \(\beta\leq 4\) is infinite. This explains the difficulty of proving the infiniteness of \(\mathcal A_k\) for \(k=1,2,3,4\).