Erdős proved that the barrier set for expProd is infinite and even has positive density.

Erdős proved that the barrier set for expProd is infinite and even has positive density.

HasPosDensity is the right reading rather than positive lower density. In [Er79d] this is Theorem 1, "the density of integers satisfying (2) is positive", where d₀(n) = ∏ αᵢ is expProd. The averaging argument there bounds the density below, but Erdős states the existence separately on the last page: "With a little more trouble, I can prove that the density of integers n for which n is a barrier for d₀(n) exists." He goes further, that if αᵢ is the density of n with max_{m<n} (m + d₀(m)) = n + i, then every αᵢ exists and they sum to 1.

[Er79d] Erdős, P., *Some unconventional problems in number theory*. Acta Math. Acad. Sci. Hungar. (1979), 71-80.