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| Factorionids (Posted on 2010-08-28) |
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A factorion is a natural number that equals the sum of the factorials of its decimal digits. There are only four factorions in base 10 and they are 1, 2, 145 and 40585.
- 1 = 1!
- 2 = 2!
- 145 = 1! + 4! + 5!
- 40585 = 4! + 0! + 5! + 8! + 5!
Let us define a factorionid as a factorion-like number which is a member of any of the factorial-like functions. Let us also define a double factorion as a factorionid of the
double factorials, a triple factorion as a factorionid of the triple factorials, and a subfactorion as a factorionid of the subfactorials.
Of these three factorionidic groups, which group has the most members? What is the largest factorionid of this group? And, of the three groups, what is the largest factorionid?
extended factorial
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 | Comment 2 of 3 | 
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we can extend the double and triple factorial to k-factorials defined as such f(0,k) = 1 f(n,k) = n for 1<=n<=k f(n,k) = n*f(n-k,k) for n>k
using this extended definition it is easy to see that for the digits 0-9 the k-factorial is identical for k>=8 so for this problem we need only consider factorians for k=1 to 8, an exhaustive search has found the following list of factorians for each k
1-factorial:
{1,2,145,40585}
2-factorial:
{1,2,3,107}
3-factorial:
{1,2,3,4,81,82,83,84}
4-factorial:
{1,2,3,4,5,49}
5-factorial:
{1,2,3,4,5,6,39}
6-factorial:
{1,2,3,4,5,6,7,29}
7-factorial:
{1,2,3,4,5,6,7,8,19}
8-factorial:
{1,2,3,4,5,6,7,8,9}
thus for k>=7 there is always 9 factorians
I am currently working on a similar extension for subfactorials
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Posted by Daniel
on 2010-08-29 02:52:05 |
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