Alternating factorial
From Wikipedia, the free encyclopedia
In mathematics, an alternating factorial is the absolute value of the alternating sum of the first n factorials.
This is the same as their sum, 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,
or with the recurrence relation
- 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)
For example, the third alternating factorial is 1! − 2! + 3!. The fourth alternating factorial is −1! + 2! - 3! + 4! = 19. Regardless of the parity of n, the last (nth) summand, n!, is given a positive sign, the (n - 1)th summand 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, since 3612703 divides af(3612702) and therefore divides af(n) for all n ≥ 3612702. As of 2006, the known primes and probable primes are af(n) for (sequence A001272 in OEIS)
- n = 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160, 661, 2653, 3069, 3943, 4053, 4998, 8275, 9158, 11164
Only the values up to n = 661 have been proved prime in 2006. af(661) is approximately 7.818097272875 × 101578.
[edit] References
- Yves Gallot, Is the number of primes
finite?
