Parental depression is a risk factor for children’s cognitive and psychological development. Literature has found reciprocal relations between parental depression and child psychopathology and effects of parental depression on children’s cognition. The present study is the first to examine reciprocity among parental depression and child cognition, and pathways to child psychopathology. Structural equation models were conducted using data from the Early Head Start Research and Evaluation Project, a nationally representative sample of 3,001 economically marginalized families. Measures were collected in four waves from 14 months to 10–11 years. Reciprocal associations emerged between maternal and paternal depression at from 14 months to 5 years. Reciprocal parental depression was associated with greater psychopathology at age 10–11. Maternal depression predicted poorer child cognition, which indirectly predicted increased depression in mothers of children aged 3–5 through paternal depression, and in fathers at age 3, through earlier paternal depression. This study was unable to parse within- and between-person effects. Additionally, data for paternal depression was limited to ages 2 and 3. Findings emphasize the transactional nature of child cognition and child and parent psychopathology, supporting family focused intervention and prevention efforts that target parent psychopathology and child cognition.

Let n and k be positive integers with $n\ge k+1$ and let $\{a_i\}_{i=1}^n$ be a strictly increasing sequence of positive integers. Let $S_{n, k}:=\sum _{i=1}^{n-k} {1}/{\mathrm {lcm}(a_{i},a_{i+k})}$ . In 1978, Borwein [‘A sum of reciprocals of least common multiples’, Canad. Math. Bull. 20 (1978), 117–118] confirmed a conjecture of Erdős by showing that $S_{n,1}\le 1-{1}/{2^{n-1}}$ . Hong [‘A sharp upper bound for the sum of reciprocals of least common multiples’, Acta Math. Hungar. 160 (2020), 360–375] improved Borwein’s upper bound to $S_{n,1}\le {a_{1}}^{-1}(1-{1}/{2^{n-1}})$ and derived optimal upper bounds for $S_{n,2}$ and $S_{n,3}$ . In this paper, we present a sharp upper bound for $S_{n,4}$ and characterise the sequences $\{a_i\}_{i=1}^n$ for which the upper bound is attained.

We prove a new irreducibility result for polynomials over ${\mathbb Q}$ and we use it to construct new infinite families of reciprocal monogenic quintinomials in ${\mathbb Z}[x]$ of degree $2^n$ .