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?
looking at the definition of the subfactorial
s(n) = n! * sum( (-1)^t/t! , t=0 to n)
the summation part is simply the series expansion of e^x for x=1
so to generalize the subfactorial to a k-subfactorial I suggest the series expansion of e^k thus
s(n,k) = n! * sum( (-1)^t * k^t / t! , t=0 to n)
thus for the digits 0 to 9 the values are for k=1 to 6:
1-subfactorial: {1,0,1,2,9,44,265,1854,14833,133496}
2-subfactorial: {1,-1,2,-2,8,8,112,656,5504,49024}
3-subfactorial: {1,-2,5,-12,33,-78,261,-360,3681,13446}
4-subfactorial: {1,-3,10,-34,120,-424,1552,-5520,21376,-69760}
5-subfactorial: {1,-4,17,-74,329,-1480,6745,-30910,143345,-663020}
6-subfactorial: {1,-5,26,-138,744,-4056,22320,-123696,690048,-3867264}
using these values I found the following k-subfactorial factorions:
1-subfactorial:
{148349}
2-subfactorial:
{2}
3-subfactorial:
{13691}
4-subfactorial:
{114}
5-subfactorial:
{}
6-subfactorial:
{21}
interesting that k=5 has no solutions
also, unlike the k-factorials, for the k-subfactorials, the range required to search increases with k.
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Posted by Daniel
on 2010-08-29 05:08:32 |
extended subfactorial
