aarjaneiro/planetmath huggingface dataset:
examples of 1-automorphic numbers

examples of 1-automorphic numbers

Concerning ourselves only with searching for

automorphic numbers

n

in bases

1

<

b

<

17

(binary to

hexadecimal

) and the ranges given by the iterator

0

<

i

<

b

3

-

1

, and limiting to

m

=

1

we find the following results:

First, it is obvious that 1 is a 1-automorphic number regardless of the base.

For the range and limit specified, there are no other 1-automorphic numbers in binary through quinary, bases 7 through 9, 11, 13 and 16.

In base 6, there are 1, 3, 4, 9, 28, 81, 136, and it is easy to verify that

3

6

2

=

13

6

,

4

6

2

=

24

6

,

13

6

2

=

213

6

,

44

6

2

=

3344

6

, etc.

In base 10, these ought to look familiar: 1, 5, 6, 25, 76, 376, 625.

Duodecimal

: 1, 4, 9, 64, 81, 513, 1216. Noticing that 4 also appears in the list for base 6, we might wonder if 4 is always 1-automorphic when

6

|

b

? The question is moot because the next

multiple

of 6 is

18

>

4

2

, thus in base 18 and any other higher bases, the square of 4 is also a 1-digit number.

Base 14: 1, 7, 8, 49, 148, 344, 2401.

Base 15: 1, 6, 10, 100, 126, 1000, 2376. Base 15 is the smallest odd base

b

to have 1-automorphic numbers in the range specified. This should not be taken to imply that it is the smallest odd base to have automorphic numbers at all.

Title

examples of 1-automorphic numbers

Canonical name

ExamplesOf1automorphicNumbers

Date of creation

2013-03-22 16:20:20

Last modified on

2013-03-22 16:20:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:21 2018 by

LaTeXML
examples of Demlo numbers

examples of Demlo numbers

In base 10, the first 20

Demlo numbers

are: 1, 121, 12321, 1234321, 123454321, 12345654321, 1234567654321, 123456787654321, 12345678987654321, 1234567900987654321, 123456790120987654321, 12345679012320987654321, 1234567901234320987654321, 123456790123454320987654321, 12345679012345654320987654321, 1234567901234567654320987654321, 123456790123456787654320987654321, 12345679012345678987654320987654321, 1234567901234567900987654320987654321, 123456790123456790120987654320987654321 (these are listed in

sequence

A002477 of Sloane’s OEIS).

In

hexadecimal

, the first few Demlo numbers are: 1, 289, 74529, 19088161, 4886709025, 1250999747361, 320255971115809, 81985529178309409, 20988295478809805601, 5373003642721911784225, 1375488932539155041567521, 352125166730061220638180129, 90144042682896272963324429089, 23076874926821455486290258903841, 5907679981266292758213173560296225, 1512366075204170948562138307930440481, etc. Written in hexadecimal, the fifteenth of these is 123456789ABCDEFEDCBA987654321. The next one is
123456789ABCDF00FEDCBA987654321. Although it is a

pandigital number

, it is not palindromic because the ”left” side is ”missing” the E that appears on the right side. As is also the case with base 10, the first non-palindromic Demlo number has two ”significant” zeroes in the middle.

Title

examples of Demlo numbers

Canonical name

ExamplesOfDemloNumbers

Date of creation

2013-03-22 16:22:09

Last modified on

2013-03-22 16:22:09

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:23 2018 by

LaTeXML
examples of digital roots in a few selected bases

examples of digital roots in a few selected bases

In base 10, the

digital roots

of the first few

integers

are: 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, … (listed in A010888 of Sloane’s OEIS), exhibiting a period of 9. Thus, all

multiples

of 9 have digital root 9 in base 10 (in fact, this is the

divisibility rule

for 9).

A prime in base 10 can have a digital root of 1, 2, 3, 4, 5, 7 or 8, or put another way, it can’t have a digital root of 6 or 9. The exclusion of 9 is not surprising, given that it is

b

-

1

. The inclusion of 3 is actually a special case: the only prime with that digital root is 3 itself. All others with digital root of 3 are multiples of 3, and the same goes for 6.

A square can have a digital root of 1, 4, 7 or 9, while a cube can have a digital root of 1, 8 or 9. Other

figurate numbers

show similar patterns in

relation

to their digital roots.

All

factorials

for

n

>

5

have digital root 9, since from that point forward they are all

divisible

by 9.

All

perfect numbers

have a digital root 1 in base 10, with the exception of 6. At least this is the case for the first 39 perfect numbers. It’s possible that if an odd perfect number exists it might have a different digital root.

A similar situation seems to hold in base 4, in which 6 has digital root 3 and the next ten or so all have a digital root of 1. They show more diversity in octal and

hexadecimal

.

In binary and

factorial base

, all

positive integers

have a digital root of 1. So, in a trivial sense, the digital root is a

multiplicative function

in those bases.

Title

examples of digital roots in a few selected bases

Canonical name

ExamplesOfDigitalRootsInAFewSelectedBases

Date of creation

2013-03-22 16:00:10

Last modified on

2013-03-22 16:00:10

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:25 2018 by

LaTeXML
examples of Kaprekar numbers

examples of Kaprekar numbers

Take the

integer

142857 in base 10. Its square is 20408122449, an 11-digit number, but for the sake of example let’s think of it as 020408122449. If we split it into two 6-digit numbers, 020408 and 122449, and add them up, we get 142857 back. Thus 142857 is a

Kaprekar number

in base 10.

The

Mersenne numbers

have plenty of base-dependent properties in binary, and one of them is that they are all Kaprekar numbers. For example, 127, which is 1111111 in binary. Its square is 11111100000001, and sure enough, 1111110 + 0000001 = 1111111.

It has been mentioned that

b

x

-

1

is a Kaprekar number in base

b

. Take 999 for example. Its square is 998001, and it’s obvious that 998 + 001 = 999. From these observations we can generalize that

(

b

x

-

1

)

2

=

(

b

x

⁢

(

b

x

-

1

)

-

1

)

+

1

.

Title

examples of Kaprekar numbers

Canonical name

ExamplesOfKaprekarNumbers

Date of creation

2013-03-22 16:02:00

Last modified on

2013-03-22 16:02:00

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:26 2018 by

LaTeXML
examples of Keith numbers

examples of Keith numbers

Take the number 47 as it is written in base 10, and start a Fibonacci-like

sequence

from its digits: 4, 7, 11, 18, 29, 47, …

42, on the other hand, isn’t a

Keith number

: 4, 2, 6, 8, 14, 22, 36, 58, etc.

For 3-digit numbers, the analogy is to the tribonacci sequence: 1, 9, 7, 17, 33, 57, 107, 197, …

In binary, 2 is a Keith number: 1, 0, 1, 1, 10, … Generalizing,

b

is a Keith number in base

b

only if it appears in the

Fibonacci sequence

, and

b

x

if in the applicable

x

bonacci sequence.

Title

examples of Keith numbers

Canonical name

ExamplesOfKeithNumbers

Date of creation

2013-03-22 16:01:35

Last modified on

2013-03-22 16:01:35

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:28 2018 by

LaTeXML
examples of palindromic primes

examples of palindromic primes

The first few

palindromic primes

in base 10 are 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727 these are listed in A002385 of Sloane’s OEIS.

In binary, all

Mersenne primes

are also palindromic primes, and the same is true of

Fermat primes

. Some other binary palindromic primes are 73, 107, 313, 443, 1193, 1453, 1571, 1619, 1787, 1831, 1879.

In

factorial base

, the most

significant digit

d

k

of a

palindromic number

has to be 1, thus a prime

p

must fall in the range

k

!

<

p

<

2

⁢

k

!

or else it is not a palindromic prime in factorial base. The first few factorial base palindromic primes are 3, 7, 11, 41, 127, 139, 173, 179, 191, 751, 811.

Title

examples of palindromic primes

Canonical name

ExamplesOfPalindromicPrimes

Date of creation

2013-03-22 15:55:33

Last modified on

2013-03-22 15:55:33

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:29 2018 by

LaTeXML
examples of parasitic numbers

examples of parasitic numbers

When using some

scientific calculators

in the course of searching for

parasitic numbers

, it might be useful to work in base 10 first, as such

calculators

might be incapable of dealing with values other than integer values in the other bases.

In base 10, the following numbers are

d

1

-parasitic (where

d

1

is the least significant digit): 1, 11, 111, 1111, 11111, 102564, 111111, 1111111, 11111111, 111111111, 1111111111, 11111111111, 102564102564, 111111111111,
1012658227848, 1111111111111, 11111111111111, 111111111111111, 1111111111111111, 11111111111111111, 102564102564102564, 105263157894736842,
111111111111111111, 1111111111111111111, 11111111111111111111,
111111111111111111111, 1014492753623188405797, 1111111111111111111111,
11111111111111111111111, 102564102564102564102564, 111111111111111111111111,
1111111111111111111111111, 11111111111111111111111111,
111111111111111111111111111, 1034482758620689655172413793,
1111111111111111111111111111, etc. In all honesty, however, the

repunits

are parasitic in a trivial sense because of the

multiplicative identity

. That’s why they’re omitted from the list of

d

1

-parasitic numbers in

sequence

A081463 of Sloane’s OEIS.

Sequence A092697 lists the smallest

d

i

-parasitic numbers for

1

<

i

<

10

: 1, 102564, 1012658227848, 105263157894736842, 1014492753623188405797, 1034482758620689655172413793, 102040816326530612244897959183673469387755, 10112359550561797752808988764044943820224719, 1016949152542372881355932203389830508474576271186440677966.

Title

examples of parasitic numbers

Canonical

name

ExamplesOfParasiticNumbers

Date of creation

2013-03-22 16:22:38

Last modified on

2013-03-22 16:22:38

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:31 2018 by

LaTeXML
examples of polydivisible numbers

examples of polydivisible numbers

Obviously, if one knows a

polydivisible number

with

k

digits then one automatically also knows

k

-

1

other polydivisible numbers.

A

consequence

of a number being polydivisible is that it’s also

divisible

by the number of digits it has. Taking

sequence

A098952 from Sloane’s OEIS and striking out: first,

odd numbers

less than 10; second, in the range

99

<

i

<

1000

, all numbers where

d

2

|̸

2

; etc., we obtain the sequence of base 10 polydivisible numbers: 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 105, 108, 120, 123, 126, 129, 132, 135, 138, 141, etc.

Title

examples of polydivisible numbers

Canonical name

ExamplesOfPolydivisibleNumbers

Date of creation

2013-03-22 16:22:41

Last modified on

2013-03-22 16:22:41

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:33 2018 by

LaTeXML
examples of self numbers in a few selected bases

examples of self numbers in a few selected bases

In base 10, the first few self numbers are 1, 3, 5, 7, 9, 20, 31, 42, 53, 64, 75, 86, 97 (listed in Sloane’s A003052). The

recurrence relation

S

i

=

8

⋅

10

i

-

1

+

S

i

-

1

+

8

with

S

1

=

9

gives the list 9, 97, 905, 8913…

In binary, the first few self numbers are 1, 4, 6, 13, 15, 18, 21, 23, 30, 32, 37, 39, 46, 48, 51, 54 (listed in A010061 of Sloane’s OEIS). The recurrence relation

S

i

=

2

k

+

S

i

-

1

+

1

with

S

1

=

1

and

k

being the number of bits in the number

S

i

-

1

has, gives the list 1, 4, 13, 30…

In

hexadecimal

, the first few self numbers are 1, 3, 5, 7, 9, 11, 13, 15, 32, 49, 66, 83, 100, 117, 134 (not currently listed in the OEIS). The recurrence relation

S

i

=

14

⋅

16

i

-

1

+

S

i

-

1

+

14

gives the list 1, 239, 3837, 61195…

In

factorial base

, the first few self numbers are 1, 4, 11, 18, 36, 43, 61, 68, 86, 93, 111, 118, 125, 132. I don’t know of a recurrence relation that will generate factorial base self numbers.

Lastly, in Roman numerals, the first few self numbers are 1, 3, 5, 7, 8, 9, 11, 13, 15.

Title

examples of self numbers in a few selected bases

Canonical name

ExamplesOfSelfNumbersInAFewSelectedBases

Date of creation

2013-03-22 15:56:18

Last modified on

2013-03-22 15:56:18

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:35 2018 by

LaTeXML
examples of Smarandache-Wellin primes in a few selected bases

examples of Smarandache-Wellin primes in a few selected bases

In base 10, there are eight known

Smarandache-Wellin primes

. The first three are 2, 23 and 2357; these are listed in A069151 of Sloane’s OEIS. The fourth is approximately

2.357111317192329313741434753

×

10

355

, and the other four are much larger.

The first three binary Smarandache-Wellin primes are 2, 11 and 751, while in

hexadecimal

the first two are 2 and 144763. There are slightly more in octal: 2, 19, 157 and 440152287140627310757403395684438501207352921. By now it should be quite obvious that 2 is a Smarandache-Wellin prime regardless of the base.

The first few Smarandache-Wellin primes in

factorial base

are 2, …

This

concept

cannot be extended to Roman numerals, as the

concatenations

yield invalid

combinations

like IIIIIV and IIIIIVVII.

Title

examples of Smarandache-Wellin primes in a few selected bases

Canonical name

ExamplesOfSmarandacheWellinPrimesInAFewSelectedBases

Date of creation

2013-03-22 15:54:43

Last modified on

2013-03-22 15:54:43

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:36 2018 by

LaTeXML
examples of trimorphic numbers

examples of trimorphic numbers

As with

examples of 1-automorphic numbers

, we shall limit ourselves to the range specified by the iterator

0

<

i

<

b

3

-

1

and the bases

1

<

b

<

17

. Again it is obvious that 1 is trimorphic regardless of the base.

It appears that a given base has more

trimorphic numbers

than 1-automorphic numbers, at least in the range observed. For binary, 1, 3, 5, 7 are trimorphic; 1, 2, 8, 26 in ternary; 1, 3, 7, 9, 15, 31, 33, 63 in quartal; 1, 4, 24, 124 in base 5. So far we’ve looked at four bases where the only 1-automorphic number in the range given is 1.

In base 6 we have: 1, 2, 3, 4, 5, 8, 9, 17, 19, 27, 28, 35, 53, 55, 80, 81, 107, 109, 135, 136, 161, 163, 189, 215, a veritable embarrasse de richesse.

Skipping ahead to base 10, it might find it useful to list only those trimorphic numbers that are not 1-automorphic: 4, 9, 24, 49, 51, 75, 99, 125, 249, 251, 375, 499 (see A056032 in Sloane’s OEIS for more of these).

Like base 6,

duodecimal

also has a lot of trimorphic numbers in the given range: 1, 3, 4, 5, 7, 8, 9, 11, 17, 55, 63, 64, 71, 73, 80, 81, 89, 127, 135, 143, 161, 351, 512, 513, 703, 863, 865, 1025, 1215, 1216, 1377, 1567, 1727.

In

hexadecimal

, the trimorphic numbers in the given range are: 1, 7, 9, 15, 127, 129, 255, 2047, 2049, 4095. In hexadecimal these all end in 1, 7, 9 or F, and if we extend our search to

b

5

-

1

, we find that this still holds true.

Title

examples of trimorphic numbers

Canonical name

ExamplesOfTrimorphicNumbers

Date of creation

2013-03-22 16:21:35

Last modified on

2013-03-22 16:21:35

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11A63

Generated on Fri Feb 9 15:22:38 2018 by

LaTeXML
extravagant number

extravagant number

An

extravagant number

or

wasteful number

n

is an

integer

with a base

b

representation

of

k

digits for which the

prime factorization

uses more than

k

digits (with repeated

prime factors

grouped with

exponents

and the digits of those exponents counted whenever greater than 1). The first few extravagant numbers in base 10 are 4, 6, 8, 9, 12, 18, 20, 22, 24, 26, 28, 30, 33, 34, 36, 38, 39, 40, 42, 44, 45, 46, 48, etc. (listed in A046760 of Sloane’s OEIS).

References

1

D. Darling, “

Economical number

” in

The Universal Book of Mathematics: From Abracadabra To Zeno’s paradoxes

. Hoboken, New Jersey: Wiley (2004)

Title

extravagant number

Canonical name

ExtravagantNumber

Date of creation

2013-03-22 16:41:20

Last modified on

2013-03-22 16:41:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Synonym

wasteful number

Related topic

FrugalNumber

Related topic

EquidigitalNumber

Generated on Fri Feb 9 15:22:45 2018 by

LaTeXML
factorial base

factorial base

A positional base in which each place value instead of being a power of the base is a

factorial

. For example, the

integer

which is represented in base 10 as 47 (because

4

⋅

10

1

+

7

⋅

10

0

) is represented in

factorial base

as 1321, or

1

⋅

4

!

+

3

⋅

3

!

+

2

⋅

2

!

+

1

⋅

1

!

.

Factorial base representation has applications in

combinatorics

and

cryptography

.

The factorial base representations are

unambiguous

as long as the maximum allowed digit for a given place value is not exceeded (e.g., the least significant digit

d

1

can only be 0 or 1, while the most significant digit in an 7-digit factorial base number

d

7

has to be in the range 0 to 7).

With this limitation placed in the definition, and the observation that

n

!

-

1

=

∑

i

=

1

n

-

1

i

!

⁢

i

it is obvious that factorial base is unambiguous, though it has the potential to use an

infinite

amount of distinct digits even as the less significant place values are limited in what values they can contain.

Though this is true of

fractions

, though in the opposite direction (the most significant fractional place values are more limited in the range of digits they can contain), factorial base has the advantage that the representation of a

rational number

always terminates. This is not always the case in a fixed base where the representation of a rational number could be repeating when the denominator is

coprime

to the base (see:

factorial base representation of fractions

).

The Lucas-Lehmer code maps unique factorial base representations of an integer

n

to the

permutation

of

n

elements

in lexicographical order.

A007623 of Sloane’s OEIS lists the first few integers written in factorial base, A046807 lists

palindromic numbers

in factorial base, A118363 lists factorial base

Harshad numbers

, etc.

Title

factorial base

Canonical name

FactorialBase

Date of creation

2013-03-22 15:55:15

Last modified on

2013-03-22 15:55:15

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Synonym

factoradic

Related topic

DecimalExpansion

Generated on Fri Feb 9 15:22:46 2018 by

LaTeXML
factorial base representation of fractions

factorial base representation of fractions

One can represent

fractions

as well as whole numbers using

factorials

much in the same way that one has, say, a decimal representation of
both whole numbers and fractions.

Suppose that

x

is a

rational number

. For simplicity, let us assume
that

0

<

x

<

1

. Then we can write

x

=

∑

k

=

2

N

d

k

k

!

where

0

≤

d

k

<

k

for some

integer

N

. Unlike decimal representations
of fractions and, more generally

representations

with any fixed base,

factorial base

representations of rational numbers all terminate.

Let us illustrate with some simple examples:

1

2

=

1

2

!

1

3

=

2

3

!

2

3

=

1

2

!

+

1

3

!

1

4

=

1

3

!

+

2

4

!

3

4

=

1

2

!

+

1

3

!

+

2

4

!

1

5

=

1

3

!

+

1

4

!

+

1

5

!

2

5

=

2

3

!

+

1

4

!

+

3

5

!

3

5

=

1

2

!

+

2

4

!

+

2

5

!

4

5

=

1

2

!

+

1

3

!

+

3

4

!

+

1

5

!

We can also employ a more concise notation as is used in
representing fractions in other bases and simply list
digits after a point. Since we would need an infinite
supply of digits, we make the same compromise as when
writing factorial base representations of integers.
With this convention, we than have the following table
of

factorial base representations of fractions

.

1/2

0 . 1

1/3

0 . 0 2

2/3

0 . 1 1

1/4

0 . 0 1 2

3/4

0 . 1 1 2

1/5

0 . 0 1 0 4

2/5

0 . 0 2 1 3

3/5

0 . 1 0 2 2

4/5

0 . 1 1 3 1

1/6

0 . 0 1

5/6

0 . 1 2

1/7

0 . 0 0 3 2 0 6

2/7

0 . 0 1 2 4 1 5

3/7

0 . 0 2 2 1 2 4

4/7

0 . 1 0 1 3 3 3

5/7

0 . 1 1 1 0 4 2

6/7

0 . 1 2 0 2 5 1

1/8

0 . 0 0 3

3/8

0 . 0 2 1

5/8

0 . 1 0 3

7/8

0 . 1 2 1

1/9

0 . 0 0 2 3 2

2/9

0 . 0 1 1 1 4

4/9

0 . 0 2 2 3 2

5/9

0 . 1 0 1 1 4

7/9

0 . 1 1 2 3 2

8/9

0 . 1 2 1 1 4

1/10

0 . 0 0 2 2

3/10

0 . 0 1 3 1

7/10

0 . 1 1 0 4

9/10

0 . 1 2 1 3

1/11

0 . 0 0 2 0 5 3 1 4 0 10

2/11

0 . 0 1 0 1 4 6 2 8 1 9

3/11

0 . 0 1 2 2 4 2 4 3 2 8

4/11

0 . 0 2 0 3 3 5 5 7 3 7

5/11

0 . 0 2 2 4 3 1 7 2 4 6

6/11

0 . 1 0 1 0 2 5 0 6 5 5

7/11

0 . 1 0 3 1 2 1 2 1 6 4

8/11

0 . 1 1 1 2 1 4 3 5 7 3

9/11

0 . 1 1 3 3 1 0 5 0 8 2

10/11

0 . 1 2 1 4 0 3 6 4 9 1

1/12

0 . 0 0 2

5/12

0 . 0 2 2

7/12

0 . 1 0 2

11/12

0 . 1 2 2

Title

factorial base representation of fractions

Canonical name

FactorialBaseRepresentationOfFractions

Date of creation

2013-03-22 16:46:01

Last modified on

2013-03-22 16:46:01

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

13

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:22:49 2018 by

LaTeXML
factorion

factorion

Given a base

b

integer

n

=

∑

i

=

1

k

d

i

⁢

b

i

-

1

where

d

1

is the least significant digit and

d

k

is the most significant, if it is also the case that

n

=

∑

i

=

1

k

d

i

!

then

n

is a

factorion

. In other words, the sum of the

factorials

of the digits in a standard positional integer base

b

(such as base 10) gives the same number as multiplying the digits by the appropriate power of that base. With the exception of 1, the

factorial base

representation of a factorion is always different from that in the integer base. Obviously, all numbers are factorions in factorial base.

1 is a factorion in any integer base. 2 is a factorion in all integer bases except binary. In base 10, there are only four factorions: 1, 2, 145 and 40585. For example,

4

×

10

4

+

0

×

10

3

+

5

×

10

2

+

8

×

10

1

+

5

×

10

0

=

4

!

+

0

!

+

5

!

+

8

!

+

5

!

=

40585

. (The factorial base representation of 40585 is 10021001).

References

1

D. Wells,

The Penguin Dictionary of Curious and Interesting Numbers

London: Penguin Group. (1987): 125

Title

factorion

Canonical

name

Factorion

Date of creation

2013-03-22 17:43:52

Last modified on

2013-03-22 17:43:52

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Classification

msc 05A10

Generated on Fri Feb 9 15:22:50 2018 by

LaTeXML