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This is just a telescoping sum: $$\sum_{n=1}^\infty \frac{n-1}{n!} = \sum_{n=1}^\infty \left(\frac{1}{(n-1)!} - \frac{1}{n!}\right) = \sum_{n=0}^\infty \frac{1}{n!} - \sum_{n=1}^\infty \frac{1}{n!} = \... • 142k 3 votes ### Given nonzero p(x)\in\mathbb Z[x]. Are there infinitely many integers n such that p(n)\mid n! is satisfied? Bober, Fretwell, Martin, and Wooley proved that when F(x)\in\mathbb Z[x] is of the form$$ F(x)=\prod_{j=1}^\ell(a_jx^{k_j}-b_j),\tag1 $$and \varepsilon>0 is arbitrary, there are infinitely ... • 7,223 1 vote ### Is there an analog for factorials in division, and if so, what are its applications and properties? As some of us have indicated in the comments to the OP, repeated division without grouping is ambiguous because division is not associative. That is,$$ (a \div b) \div c \not= a \div (b \div c)  ...
I used such approach: for given $n$ (currently, $n=32$), loop through prime numbers $p$ starting from certain value $p_0$ to, theoretically, $\sqrt{\sum_{k=1}^n k!}$; and for these $(n,p)$ construct 2 ...