alternating factorial

The alternating factorialMathworldPlanetmath af(n) of a positive integer n is the sum


which can also be expressed with the recurrence relation af(n)=n!-af(n-1) with starting condition af(1)=1. The notation n¡! (alternating an inverted exclamation mark with a regular exclamation mark) has been proposed by analogy to that of the double factorialMathworldPlanetmath, but has not gained much support, in part because of TeX’s lack of support for Spanish charactersMathworldPlanetmathPlanetmath.

The first few alternating factorials, listed in A005165 of Sloane’s OEIS, are 1, 5, 19, 101, 619, 4421.

In 1999, Miodrag Zivković proved that gcd(n,af(n))=1 and that the set of alternating factorials that are prime numbersMathworldPlanetmath is finite. af(661) is the largest such known prime.

Title alternating factorial
Canonical name AlternatingFactorial
Date of creation 2013-03-22 16:19:59
Last modified on 2013-03-22 16:19:59
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 7
Author PrimeFan (13766)
Entry type Definition
Classification msc 05A10