On the distribution of odd values of 2a-regular partition functions

Let b2a(n) denote the number of 2a-regular partitions of n. Suppose j is a positive integer and i is odd. If a∈{1,2}, we show that#{0≤n≤X:b2a(n)≡i(mod2j)}≫jX. This improves a result of Ono and Penniston [6]. For a≥3 odd, we show that#{0≤n≤X:b2a(n)≡i(mod2j)}≫a,jXlogX. Finally for a≥4 even, we prove that#{0≤n≤X:b2a(n)≡i(mod2j)}≫a,jXlogX(loglogX)δ(a), where δ(a)=2a2−1−2.