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1
D e f i n i t i o n
2
F a c t o r i o n s f o r S F D b
T o g g l e F a c t o r i o n s f o r S F D b s u b s e c t i o n
2 . 1
b = ( k − 1 ) !
2 . 2
b = k ! − k + 1
2 . 3
T a b l e o f f a c t o r i o n s a n d c y c l e s o f S F D b
3
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4
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5
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A p p e a r a n c e
F r o m W i k i p e d i a , t h e f r e e e n c y c l o p e d i a
Let
n
{\displaystyle n}
be a natural number. For a base
b
>
1
{\displaystyle b>1}
, we define the sum of the factorials of the digits [5] [6] of
n
{\displaystyle n}
,
SFD
b
:
N
→
N
{\displaystyle \operatorname {SFD} _{b}:\mathbb {N} \rightarrow \mathbb {N} }
, to be the following:
SFD
b
(
n
)
=
∑
i
=
0
k
−
1
d
i
!
.
{\displaystyle \operatorname {SFD} _{b}(n )=\sum _{i=0}^{k-1}d_{i}!.}
where
k
=
⌊
log
b
n
⌋
+
1
{\displaystyle k=\lfloor \log _{b}n\rfloor +1}
is the number of digits in the number in base
b
{\displaystyle b}
,
n
!
{\displaystyle n!}
is the factorial of
n
{\displaystyle n}
and
d
i
=
n
mod
b
i
+
1
−
n
mod
b
i
b
i
{\displaystyle d_{i}={\frac {n{\bmod {b^{i+1}}}-n{\bmod {b^{i}}}}{b^{i}}}}
is the value of the
i
{\displaystyle i}
th digit of the number. A natural number
n
{\displaystyle n}
is a
b
{\displaystyle b}
-factorion if it is a fixed point for
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
, i.e. if
SFD
b
(
n
)
=
n
{\displaystyle \operatorname {SFD} _{b}(n )=n}
.[7]
1
{\displaystyle 1}
and
2
{\displaystyle 2}
are fixed points for all bases
b
{\displaystyle b}
, and thus are trivial factorions for all
b
{\displaystyle b}
, and all other factorions are nontrivial factorions .
For example, the number 145 in base
b
=
10
{\displaystyle b=10}
is a factorion because
145
=
1
!
+
4
!
+
5
!
{\displaystyle 145=1!+4!+5!}
.
For
b
=
2
{\displaystyle b=2}
, the sum of the factorials of the digits is simply the number of digits
k
{\displaystyle k}
in the base 2 representation since
0
!
=
1
!
=
1
{\displaystyle 0!=1!=1}
.
A natural number
n
{\displaystyle n}
is a sociable factorion if it is a periodic point for
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
, where
SFD
b
k
(
n
)
=
n
{\displaystyle \operatorname {SFD} _{b}^{k}(n )=n}
for a positive integer
k
{\displaystyle k}
, and forms a cycle of period
k
{\displaystyle k}
. A factorion is a sociable factorion with
k
=
1
{\displaystyle k=1}
, and a amicable factorion is a sociable factorion with
k
=
2
{\displaystyle k=2}
.[8] [9]
All natural numbers
n
{\displaystyle n}
are preperiodic points for
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
, regardless of the base. This is because all natural numbers of base
b
{\displaystyle b}
with
k
{\displaystyle k}
digits satisfy
b
k
−
1
≤
n
≤
(
b
−
1
)
!
(
k
)
{\displaystyle b^{k-1}\leq n\leq (b-1)!(k )}
. However, when
k
≥
b
{\displaystyle k\geq b}
, then
b
k
−
1
>
(
b
−
1
)
!
(
k
)
{\displaystyle b^{k-1}>(b-1)!(k )}
for
b
>
2
{\displaystyle b>2}
, so any
n
{\displaystyle n}
will satisfy
n
>
SFD
b
(
n
)
{\displaystyle n>\operatorname {SFD} _{b}(n )}
until
n
<
b
b
{\displaystyle n<b^{b}}
. There are finitely many natural numbers less than
b
b
{\displaystyle b^{b}}
, so the number is guaranteed to reach a periodic point or a fixed point less than
b
b
{\displaystyle b^{b}}
, making it a preperiodic point. For
b
=
2
{\displaystyle b=2}
, the number of digits
k
≤
n
{\displaystyle k\leq n}
for any number, once again, making it a preperiodic point. This means also that there are a finite number of factorions and cycles for any given base
b
{\displaystyle b}
.
The number of iterations
i
{\displaystyle i}
needed for
SFD
b
i
(
n
)
{\displaystyle \operatorname {SFD} _{b}^{i}(n )}
to reach a fixed point is the
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
function's persistence of
n
{\displaystyle n}
, and undefined if it never reaches a fixed point.
Factorions for SFDb [ edit ]
b = (k − 1)![ edit ]
Let
k
{\displaystyle k}
be a positive integer and the number base
b
=
(
k
−
1
)
!
{\displaystyle b=(k-1)!}
. Then:
n
1
=
k
b
+
1
{\displaystyle n_{1}=kb+1}
is a factorion for
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
for all
k
.
{\displaystyle k.}
SFD
b
(
n
1
)
=
d
1
!
+
d
0
!
{\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}
=
k
!
+
1
!
{\displaystyle =k!+1!}
=
k
(
k
−
1
)
!
+
1
{\displaystyle =k(k-1)!+1}
=
d
1
b
+
d
0
{\displaystyle =d_{1}b+d_{0}}
=
n
1
{\displaystyle =n_{1}}
Thus
n
1
{\displaystyle n_{1}}
is a factorion for
F
b
{\displaystyle F_{b}}
for all
k
{\displaystyle k}
.
n
2
=
k
b
+
2
{\displaystyle n_{2}=kb+2}
is a factorion for
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
for all
k
{\displaystyle k}
.
SFD
b
(
n
2
)
=
d
1
!
+
d
0
!
{\displaystyle \operatorname {SFD} _{b}(n_{2})=d_{1}!+d_{0}!}
=
k
!
+
2
!
{\displaystyle =k!+2!}
=
k
(
k
−
1
)
!
+
2
{\displaystyle =k(k-1)!+2}
=
d
1
b
+
d
0
{\displaystyle =d_{1}b+d_{0}}
=
n
2
{\displaystyle =n_{2}}
Thus
n
2
{\displaystyle n_{2}}
is a factorion for
F
b
{\displaystyle F_{b}}
for all
k
{\displaystyle k}
.
Factorions
k
{\displaystyle k}
b
{\displaystyle b}
n
1
{\displaystyle n_{1}}
n
2
{\displaystyle n_{2}}
4
6
41
42
5
24
51
52
6
120
61
62
7
720
71
72
b = k ! − k + 1[ edit ]
Let
k
{\displaystyle k}
be a positive integer and the number base
b
=
k
!
−
k
+
1
{\displaystyle b=k!-k+1}
. Then:
n
1
=
b
+
k
{\displaystyle n_{1}=b+k}
is a factorion for
SFD
b
{\displaystyle \operatorname {SFD} _{b}}
for all
k
{\displaystyle k}
.
SFD
b
(
n
1
)
=
d
1
!
+
d
0
!
{\displaystyle \operatorname {SFD} _{b}(n_{1})=d_{1}!+d_{0}!}
=
1
!
+
k
!
{\displaystyle =1!+k!}
=
k
!
+
1
−
k
+
k
{\displaystyle =k!+1-k+k}
=
1
(
k
!
−
k
+
1
)
+
k
{\displaystyle =1(k!-k+1)+k}
=
d
1
b
+
d
0
{\displaystyle =d_{1}b+d_{0}}
=
n
1
{\displaystyle =n_{1}}
Thus
n
1
{\displaystyle n_{1}}
is a factorion for
F
b
{\displaystyle F_{b}}
for all
k
{\displaystyle k}
.
Factorions
k
{\displaystyle k}
b
{\displaystyle b}
n
1
{\displaystyle n_{1}}
3
4
13
4
21
14
5
116
15
6
715
16
Table of factorions and cycles of SFDb [ edit ]
All numbers are represented in base
b
{\displaystyle b}
.
See also [ edit ]
References [ edit ]
^ Madachy, Joseph S. (1979), Madachy's Mathematical Recreations , Dover Publications, p. 167, ISBN 9780486237626
^ Pickover, Clifford A. (1995), "The Loneliness of the Factorions", Keys to Infinity , John Wiley & Sons, pp. 169–171 and 319–320, ISBN 9780471193340 – via Google Books
^ Gupta, Shyam S. (2004), "Sum of the Factorials of the Digits of Integers", The Mathematical Gazette , 88 (512), The Mathematical Association: 258–261, doi :10.1017/S0025557200174996 , JSTOR 3620841 , S2CID 125854033
^ Sloane, Neil, "A061602" , On-Line Encyclopedia of Integer Sequences
^ Abbott, Steve (2004), "SFD Chains and Factorion Cycles", The Mathematical Gazette , 88 (512), The Mathematical Association: 261–263, doi :10.1017/S002555720017500X , JSTOR 3620842 , S2CID 99976100
^ a b Sloane, Neil, "A214285" , On-Line Encyclopedia of Integer Sequences
^ a b Sloane, Neil, "A254499" , On-Line Encyclopedia of Integer Sequences
^ Sloane, Neil, "A193163" , On-Line Encyclopedia of Integer Sequences
External links [ edit ]
t
e
Possessing a specific set of other numbers
Expressible via specific sums
R e t r i e v e d f r o m " https://en.wikipedia.org/w/index.php?title=Factorion&oldid=1169980467 "
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