An alternating factorial is the sum of the first n factorials with the odd-indexed factorials multiplied by −1 if n is even, and the even-indexed factorials multiplied by −1 if n is odd, resulting in an alternation of signs of the summands (or alternation of addition and subtraction operators, if preferred). To put it algebraically, In mathematics, the factorial of a natural number n is the product of all positive integers less than and equal to n. ... In mathematics, −1 is the integer greater than negative two (−2) and less than 0. ...

or with the recurrence relation Recurrent redirects here; for the meaning of recurrent in contemporary hit radio, see Recurrent rotation. ...

af(n) = n! − af(n − 1)

in which af(1) = 1.

The first few alternating factorials are

1, 1, 5, 19, 101, 619, 4421, 35899, 326981, 3301819, 36614981, 442386619, 5784634181, 81393657019 (sequence A005165 in OEIS) Look up one in Wiktionary, the free dictionary. ... 5 (five) is a number, numeral, and glyph. ... 19 (nineteen) is the natural number following 18 and preceding 20. ... 101 (one hundred [and] one) is the natural number following 100 and preceding 102. ... The On-Line Encyclopedia of Integer Sequences (OEIS) is a web-based searchable database of integer sequences. ...

For example, the third alternating factorial is 1! + −(2!) + 3! = 5, or if preferred, 1! − 2! + 3! The fourth alternating factorial is −(1!) + 2! + −(3!) + 4! = 19. Regardless of the parity of n, the summand n − 1 is given a negative sign and the signs of the lower-indexed summands are alternated accordingly.

This pattern of alternation ensures the resulting sums are all positive integers. Changing the rule so that either the odd- or even-indexed summands are given negative signs (regardless of the parity of n) changes the signs of the resulting sums but not their absolute values.

Except for n = 1, the factorial of n and the alternating factorial of n are coprime. Miodrag Zivković proved in 1999 that there are only a finite number of alternating factorials that are also prime numbers. The largest alternating factorial known to be a prime is the alternating factorial of 661, approximately 7.818097272875 × 10¹⁵⁷⁸. Coprime - Wikipedia /**/ @import /skins-1. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ...

Reference

• Yves Gallot, Is the number of primes finite?