For integers $n\geq 1$ let $G_n$ be the Gregory coefficients or reciprocal logarithmic numbers, see the Wikipedia [Gregory coefficients.] (https://en.wikipedia.org/wiki/Gregory_coefficients) $${z\over\log(1+z)}=1+\sum_1^{\infty}G_nz^n$$ I think that I was inspired in [1] to ask the question.

Question. Let $1\leq N<M$ where $N$ and $M$ are integers. Is it possible to determine examples of a choice of integers $0\leq \alpha,\beta,\gamma$ for which $$\sum_{k=N}^M k^{\alpha}2^{\beta k}k!^{\gamma}|G_k|$$ is never an integer for all $N$ and $M$? Many thanks.

I suspect of some simple example from a few computations that I did with a Pari/GP program. I would like to know some case(s) with a good mathematical content. In your examples you can take some of those exponents $\alpha$ or $\beta$ or $\gamma$ equals to zero, but the case $\alpha=\beta=\gamma=0$ is excluded in this disccussion.

References:

[1] Thomas J. Osler, Partial sums of series that cannot be an integer, The Mathematical Gazette Vol. 96, No. 537 (November 2012), pp. 515-519.