Understanding about paper related to integer partition with even parts distinct

Let $$ped(n)$$ denote the number of partition of a positive integer $$n$$ wherein the even parts are distinct and the odd parts are unrestricted. I follow this paper:

Andrews, G. E., Hirschhorn, M. D., & Sellers, J. A. (2010). Arithmetic properties of partitions with even parts distinct. The Ramanujan Journal, 23(1-3), 169-181.

and found some confusions there. They are:

1. $$ped(n)$$ is divisible by $$6$$ at least $$\frac16$$ of the time. What is the meaning of "at least..."? Is it such a probability or what?
2. The proof by induction of third congruence in the Theorem 3.4. : $$ped(3^{2\alpha}n+\frac{3^{2\alpha}-1}{8})q^n \equiv (-1)^{\alpha-1}\psi(-q)\varphi(-q^3) \pmod3$$.

Anyone can help me to understanding these, especially about number 2? Many thanks in advanced.

1. Their result about $$ped(9k+7)$$ and Corollary 3.6 mean they can guarantee $$1/6$$ of $$ped(n)$$ values are divisible by 6. There are certainly other values of $$n$$ for which $$ped(n)$$ is a multiple of 6, thus the "at least" statement. Compare Ramanujan's result that $$p(5k+4) \equiv 0 \bmod 5$$ for every $$k$$. That doesn't give every value for which $$p(n) \equiv 0 \bmod 5$$, e.g., $$p(7) = 15$$. So Ramanujan's result can be stated as "at least 1/5 of $$p(n)$$ values are multiples of 5."
2. The $$ped(9n+1)$$ result is the $$\alpha=1$$ base case. A related paper from the same journal in the same year goes through a very similar induction argument in detail: Look at Hirschhorn and Sellers, Arithmetic Properties of Partitions with Odd Parts Distinct, Ramanujan Journal 22(3) (2010) 273-284.
Edit: More detail for #1. Knowing $$ped(9k+7) \equiv 0 \bmod 6$$ for every nonnegative $$k$$ establishes that $$ped(n)$$ is a multiple of 6 for at least 1/9 of all values of $$n$$. The $$\alpha = 1$$ cases of Corollary 3.6 say $$ped(27k + 19)$$ and $$ped(81k + 64)$$ are multiples of 6, giving another 1/27 and 1/81 of all values of $$n$$ (one should confirm that none of these sequences overlap). The $$\alpha = 2$$ cases of Corollary 3.6 say $$ped(243k + 172)$$ and $$ped(729k + 577)$$ are multiples of 6, giving another 1/243 and 1/729 of all values of $$n$$, etc. Thus the partial geometric series 1/9 + 1/27 + 1/81 + 1/243 + ...
• I mean, where did the numbers $1/9+1/27+...=1/6$ came from? Apr 27 at 6:01