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    Packing Characteristics of Different Shaped Proppants for use with Hydrofracing - A Numerical Investigation using 3D FEMDEM

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    Unlimited parity alternating partitions

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    We introduce a new type of partitions that consists of partitions whose different parts alternate in parity (e.g., 3+2+2+1+13+2+2+1+1). Various properties of this partition function are studied. In particular, we obtain its asymptotic behavior by employing Ingham's Tauberian theorem

    Note on sums involving the Euler function

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    In this note, we provide refined estimates of the following sums involving the Euler totient function: βˆ‘n≀xΟ•([xn])andβˆ‘n≀xΟ•([x/n])[x/n]\sum_{n\le x} \phi\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{\phi([x/n])}{[x/n]} where [x][x] denotes the integral part of real xx. The above summations were recently considered by Bordell\`es et al. and Wu.Comment: Bull. Aust. Math. Soc., accepte
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