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An alternative notation for the rising factorial () is the less common () +. When () + is used to denote the rising factorial, the notation () is typically used for the ordinary falling factorial, to avoid confusion. [3]
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For =, the sum of the factorials of the digits is simply the number of digits in the base 2 representation since ! =! =. A natural number n {\displaystyle n} is a sociable factorion if it is a periodic point for SFD b {\displaystyle \operatorname {SFD} _{b}} , where SFD b k ( n ) = n {\displaystyle \operatorname {SFD} _{b}^{k}(n)=n} for a ...
SymPy is an open-source Python library for symbolic computation.It provides computer algebra capabilities either as a standalone application, as a library to other applications, or live on the web as SymPy Live [2] or SymPy Gamma. [3]
Since ! is the product of the integers 1 through n, we obtain at least one factor of p in ! for each multiple of p in {,, …,}, of which there are ⌊ ⌋.Each multiple of contributes an additional factor of p, each multiple of contributes yet another factor of p, etc. Adding up the number of these factors gives the infinite sum for (!
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The hyperfactorials were studied beginning in the 19th century by Hermann Kinkelin [3] [4] and James Whitbread Lee Glaisher. [5] [4] As Kinkelin showed, just as the factorials can be continuously interpolated by the gamma function, the hyperfactorials can be continuously interpolated by the K-function.