aarjaneiro/planetmath huggingface dataset:
Mian-Chowla sequence

Mian-Chowla sequence

The

Mian-Chowla sequence

is a

B

2

sequence

with

a

1

=

1

and

a

n

for

n

>

2

being the smallest

integer

such that each pairwise sum

a

i

+

a

j

is distinct, where

0

<

i

<

(

n

+

1

)

and likewise for

j

, that is,

1

≤

i

≤

j

≤

n

. The case

i

=

j

is always considered.

At the beginning, with

a

1

, there is only one pairwise sum, 2.

a

2

can be 2 since the pairwise sums then are 2, 3 and 4.

a

3

can’t be 3 because then there would be the pairwise sums 1 + 3 = 2 + 2 = 4. Thus

a

3

=

4

. The sequence, listed in A005282 of Sloane’s OEIS, continues 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, 401, 475, … If we define

a

1

=

0

, the resulting sequence is the same except each term is one less.

Rachel Lewis noticed that

∑

i

=

1

∞

1

a

i

≡

2.1585

, a constant listed in Finch’s book.

One way to calculate the Mian-Chowla sequence in Mathematica is thus:

a = Table[1, {40}];
n = 2;
test = 1;
While[n < 41,
mcFlag = False;
While[Not[mcFlag],
test++;
a[[n]] = test;
pairSums = Flatten[Table[a[[i]] + a[[j]], {i, n}, {j, i, n}]];
mcFlag = TrueQ[Length[pairSums] == Length[Union[pairSums]]]
];
n++
];
a

References

1

S. R. Finch,

Mathematical Constants

, Cambridge (2003): Section 2.20.2

2

R. K. Guy

Unsolved Problems in

Number Theory

, New York: Springer (2003)

Title

Mian-Chowla sequence

Canonical

name

MianChowlaSequence

Date of creation

2013-03-22 16:27:49

Last modified on

2013-03-22 16:27:49

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B13

Generated on Fri Feb 9 15:26:10 2018 by

LaTeXML
Schnirelmann density

Schnirelmann density

Let

A

be a subset of

ℤ

, and let

A

⁢

(

n

)

be number of elements of

A

in

[

1

,

n

]

.

of

A

is

σ

⁢

A

=

inf

n

⁡

A

⁢

(

n

)

n

.

has the following properties:

1.

A

⁢

(

n

)

≥

n

⁢

σ

⁢

A

for all

n

.

2.

σ

⁢

A

=

1

if and only if

ℕ

⊆

A

3.

if

1

does not belong to

A

, then

σ

⁢

A

=

0

.

Schnirelmann proved that if

0

∈

A

∩

B

then

σ

⁢

(

A

+

B

)

≥

σ

⁢

A

+

σ

⁢

B

-

σ

⁢

A

⋅

σ

⁢

B

and also if

σ

⁢

A

+

σ

⁢

B

≥

1

, then

σ

⁢

(

A

+

B

)

=

1

. From these he deduced that if

σ

⁢

A

>

0

then

A

is an

additive basis

.

Title

Schnirelmann density

Canonical name

SchnirelmannDensity

Date of creation

2013-03-22 13:19:36

Last modified on

2013-03-22 13:19:36

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

9

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B13

Classification

msc 11B05

Synonym

Shnirel’man density

Synonym

Shnirelman density

Related topic

Basis2

Related topic

EssentialComponent

Related topic

MannsTheorem

Generated on Fri Feb 9 15:26:12 2018 by

LaTeXML
sumset

sumset

Let

A

1

,

A

2

,

…

,

A

n

be subsets of an

additive group

G

. The sumset

A

1

+

A

2

+

⋯

+

A

n

is the set of all elements of the form

a

1

+

a

2

+

⋯

+

a

n

, where

a

i

∈

A

i

.

In geometry a sumset is often called

Minkowski sum

.

Title

sumset

Canonical name

Sumset

Date of creation

2013-03-22 13:19:14

Last modified on

2013-03-22 13:19:14

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

5

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B13

Classification

msc 11B05

Synonym

Minkowski sum

Related topic

MinkowskiSum2

Generated on Fri Feb 9 15:26:14 2018 by

LaTeXML
Ulam number

Ulam number

The

n

th

Ulam number

U

n

for

n

>

2

is the smallest number greater than

U

n

-

1

which is a sum of two smaller Ulam numbers in a unique way.

U

1

=

1

and

U

2

=

2

; the

sequence

continues 3, 4, 6, 8, 11, 13, 16, 18, 26, 28, 36, 38, 47, 48, 53, 57, 62, 69, 72, 77, 82, 87, 97, 99, 102, 106, 114, 126, 131, 138, 145, 148, 155, 175, 177, 180, 182, 189, 197, etc. (listed in A002858 of Sloane’s OEIS); it is what is usually referred to as the

Ulam sequence

.

So, for example, 47 is an Ulam number because it is the sum of the pair of smaller Ulam numbers 11 and 36, and no other pair, while 48 is also an Ulam number because it is the sum of 1 and 47, and no other pair. 49 is not an Ulam number because it is the sum of 1 and 48, and of 2 and 47.

Stanisław Ulam (

http://planetmath.org/StanislawUlam

) first studied this sequences in the 1960s “in a peculiar attempt to get a 1D analog of a 2D

cellular automaton

” (Wolfram, 2002). In 2001 Jud McCranie verified that among the first 40000000, the only consecutive pairs that are also both Ulam numbers are 1 and 2, 2 and 3, 3 and 4, and 47 and 48. More recently, in 2006, Neil Sloane conjectured that a plot of the Ulam numbers will produce a line that is very close to

y

=

1351

100

⁢

x

.

Ulam numbers and the resulting Ulam sequences can be generalized to having different initial values

U

1

and

U

2

with the only requirement being that

U

1

<

U

2

, these are sometimes referred to as

Ulam-type sequences

. If

U

1

=

2

and

2

∤

U

2

, then the Ulam-type sequence will have only one other even term (Schmerl & Spiegel, 1994).

References

1

J. Schmerl & E. Spiegel, “The Regularity of Some 1-Additive Sequences”.

J. Combinatoric Theory Ser.

A

66

(1994): 172 - 175

2

S. Wolfram

A New Kind of Science

New York: Wolfram Media (2002): 908

Title

Ulam number

Canonical

name

UlamNumber

Date of creation

2013-03-22 16:46:24

Last modified on

2013-03-22 16:46:24

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B13

Defines

Ulam sequence

Defines

Ulam-type sequence

Generated on Fri Feb 9 15:26:16 2018 by

LaTeXML
addition

addition

Addition

is a mathematical

operation

in which two or more numbers are added up. The numbers being added are usually called the

addends

, while the result is usually called the

sum

. The numbers may be real,

imaginary

or complex. Two examples: 2 + 2 = 4;

4

+

π

≈

7.14159

. In the realm of

real numbers

, adding up positive numbers yields a result that is greater than any of the summands.

The usual symbol for addition is the cross with its four arms of equal length pointing north, east, west and south: +. This

operator

is used in standard

infix notation

as well as in

Polish notation

and

reverse Polish notation

. The operation of addition is

commutative

, so

x

+

y

=

y

+

z

. Moreover, it is associative, so

x

+

(

y

+

z

)

=

(

x

+

y

)

+

z

.
Combining these two facts leads to the observation that if

S

is a

finite set

of numbers, then we can add them up one by one in any order without any potential ambiguity in the resulting sum. This allows us to define the

sigma notation

∑

x

∈

S

x

for the sum of finitely many numbers. For example, the sum of the first

n

positive

perfect

cubes can be written as

∑

i

∈

{

1

,

…

,

n

}

i

3

.

An alternate way of writing the same sum is

∑

i

=

1

n

i

3

.

Here we might imagine the sum as being produced by an iterative process. But because of the aforementioned commutativity and associativity, it doesn’t matter if the

iterator

is started at the smallest value or the largest value.

A notable

property

of the iterative sum is that

∑

i

=

1

n

i

3

=

(

∑

i

=

1

n

i

)

2

.

(See Nicomachus’

theorem

).

Besides the possibility of overflow, addition presents no problems for

fixed point arithmetic

provided the operands are representable in

fixed point

to begin with. In floating point there exists the possibility of loss of precision (if for example we were to add up several different

irrational numbers

).

Title

addition

Canonical

name

Addition

Date of creation

2013-03-22 16:35:28

Last modified on

2013-03-22 16:35:28

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B25

Classification

msc 00A06

Classification

msc 00A05

Related topic

Summation

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

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alternating sum

alternating sum

An

alternating sum

is a

sequence

of

arithmetic operations

in which each

addition

is followed by a

subtraction

, and viceversa, applied to a sequence of numerical entities. For example,

log

⁡

2

=

1

-

1

2

+

1

3

-

1

4

+

1

5

-

1

6

+

1

7

-

…

An alternating sum is also called an alternating series.

Alternating sums are often expressed in

summation

notation with the iterated

expression

involving

multiplication

by

negative

one raised to the iterator. Since a

negative number

raised to an

odd number

gives a negative number while raised to an

even number

gives a positive number (see:

factors with minus sign

),

(

-

1

)

i

essentially has the effect of turning the odd-indexed terms of the sequence negative but keeping their

absolute values

the same. Our previous example would thus be restated

log

⁡

2

=

∑

i

=

1

∞

(

-

1

)

i

-

1

⁢

1

i

.

If the operands in an alternating sum decrease in value as the iterator increases, and approach zero, then the alternating sum

converges

to a specific value. This fact is used in many of the best-known expression for

π

or

fractions

thereof, such as the

Gregory series

:

π

4

=

∑

i

=

0

∞

(

-

1

)

i

⁢

1

2

⁢

i

+

1

Other

constants

also find expression as alternating sums, such as Cahen’s constant.

An alternating sum need not necessarily involve an

infinity

of operands. For example, the

alternating factorial

of

n

is computed by an alternating sum stopping at

i

=

n

.

References

1

Tobias Dantzig,

Number: The

Language

of Science

, ed. Joseph Mazur. New York: Pi Press (2005): 166

Title

alternating sum

Canonical

name

AlternatingSum

Date of creation

2013-03-22 17:35:30

Last modified on

2013-03-22 17:35:30

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B25

Generated on Fri Feb 9 15:26:24 2018 by

LaTeXML
covering system

covering system

A

covering system

is a system of

congruences

such that every

natural number

n

is “covered” by at least one of the congruences, that is, given the

finite sets

a

and

m

both containing

k

nonnegative integers

(but each

m

i

>

1

), for every

n

there’s at least one value of

i

such that

n

≡

a

i

mod

m

i

.

For example, Davenport gives the following system:

0

mod

2

,

0

mod

3

,

1

mod

4

,

1

mod

6

and

11

mod

12

. For the sake of demonstration it will be

sufficient

here to show that this system of congruences covers all

12

<

n

<

37

. Examining the congruences in the order stated for the first half of our sample range, the

multiples

of 2 and 3 are covered by the first two congruences, leaving us just 13, 17, 19 and 23 to worry about. The third congruence takes care of 13 and 17. The fourth congruence takes care of 19, with 13 already taken care of. The fifth congruence covers 23. In many cases, a particular number will be taken care of by more than one congruence. The following table shows all the congruences that cover the

remainder

of our sample range.

25

1

mod

4

1

mod

6

26

0

mod

2

27

0

mod

3

28

0

mod

2

29

1

mod

4

30

0

mod

2

0

mod

3

31

1

mod

6

32

0

mod

2

33

0

mod

3

1

mod

4

34

0

mod

2

35

11

mod

12

36

0

mod

2

0

mod

3

There are various

open problems

pertaining to covering systems. Erdős conjectured that for any

positive

N

there is always a covering system in which all the moduli are greater than

N

. Another one, posed by Erdős and Selfridge is whether there is a covering system in which all the moduli are odd. Erdős has presented a covering system which does not use 2 as a

modulus

but it does use 4.

References

1

H. Davenport,

The

Higher Arithmetic

, Sixth Edition. Cambridge: Cambridge University Press (1995): 57 - 58

2

Paul Erdős & János Surányi

Topics in the theory of numbers

New York: Springer (2003): 46

Title

covering system

Canonical

name

CoveringSystem

Date of creation

2013-03-22 18:04:37

Last modified on

2013-03-22 18:04:37

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

7

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B25

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

LaTeXML
Freiman’s theorem

Freiman’s theorem

Let

A

be a

finite set

of

integers

such that the

2

-fold sumset

2

⁢

A

is “small”, i.e.,

|

2

⁢

A

|

<

c

⁢

|

A

|

for some

constant

c

. There exists an

n

-dimensional arithmetic progression (

http://planetmath.org/MulidimensionalArithmeticProgression

) of length

c

′

⁢

|

A

|

that contains

A

, and such that

c

′

and

n

are

functions

of

c

only.

References

1

Melvyn B. Nathanson.

Additive Number Theory

: Inverse Problems and Geometry of
Sumsets

, volume 165 of

GTM

.

Springer, 1996.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003

Zbl 0859.11003.

Title

Freiman’s

theorem

Canonical name

FreimansTheorem

Date of creation

2013-03-22 13:39:05

Last modified on

2013-03-22 13:39:05

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11B25

Generated on Fri Feb 9 15:26:28 2018 by

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how to use a table

how to use a table

In this entry, how to use a table to multiply numbers will be explained and illustrated with examples. The general rule is as follows:

Suppose you want to multiply two numbers. Find the first number
in the very first column. Follow the row in which this number appears
until you land in the column which is labelled with the second number on
top. The number on which you land is your answer.

Let us illustrate with an example. To keep things simple, we will
use a 5 by 5 multiplication table

×

1

2

3

4

5

1

1

2

3

4

5

2

2

4

6

8

10

3

3

6

9

12

15

4

4

8

12

16

20

5

5

10

15

20

25

Suppose we want to figure out

3

×

4

. We first find the
row which starts with “3”. That row is highlighted below:

×

1

2

3

4

5

1

1

2

3

4

5

2

2

4

6

8

10

3

3

6

9

12

15

4

4

8

12

16

20

5

5

10

15

20

25

Next we follow this row until we find the column which has “4”
on top. This column is highlighted below:

×

1

2

3

4

5

1

1

2

3

4

5

2

2

4

6

8

10

3

3

6

9

12

15

4

4

8

12

16

20

5

5

10

15

20

25

Note that “12” appears where the highlighted row and the highlighted
column cross. (For convenience, it has been underlined.) Therefore,
we found that

3

×

4

=

12

.

Title

how to use a table

Canonical name

HowToUseATable

Date of creation

2013-03-22 16:36:25

Last modified on

2013-03-22 16:36:25

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Example

Classification

msc 11B25

Generated on Fri Feb 9 15:26:29 2018 by

LaTeXML
multidimensional arithmetic progression

multidimensional arithmetic progression

An

n

-dimensional

arithmetic

progresssion is a set of the form

Q

=

Q

⁢

(

a

;

q

1

,

…

,

q

n

;

l

1

,

…

,

l

n

)

=

{

a

+

x

1

⁢

q

1

+

⋯

+

x

n

⁢

q

n

∣

0

≤

x

i

<

l

i

⁢

for

⁢

i

=

1

,

…

,

n

}

.

The length of the progression is defined as

l

1

⁢

⋯

⁢

l

n

. The progression is

proper

if

|

Q

|

=

l

1

⁢

⋯

⁢

l

n

.

References

1

Melvyn B. Nathanson.

Additive Number Theory

:

Inverse Problems

and

Geometry

of
Sumsets

, volume 165 of

GTM

.

Springer, 1996.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0859.11003

Zbl 0859.11003.

Title

multidimensional arithmetic progression

Canonical

name

MultidimensionalArithmeticProgression

Date of creation

2013-03-22 13:39:02

Last modified on

2013-03-22 13:39:02

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

7

Author

bbukh (348)

Entry type

Definition

Classification

msc 11B25

Synonym

generalized arithmetic progression

Related topic

ArithmeticProgression

Generated on Fri Feb 9 15:26:31 2018 by

LaTeXML
Szemerédi’s theorem

Szemerédi’s theorem

Let

k

be a

positive integer

and let

δ

>

0

. There exists a positive integer

N

=

N

⁢

(

k

,

δ

)

such that every subset of

{

1

,

2

,

…

,

N

}

of size (

http://

planetmath

.org/

Cardinality

)

δ

⁢

N

contains an

arithmetic progression

of length

k

.

The case

k

=

3

was first proved by Roth

[

4

]

. His method did not seem to extend to the case

k

>

3

. Using completely different ideas Szemerédi proved the case

k

=

4

[

5

]

, and the general case of an arbitrary

k

[

6

]

.

The best known bounds for

N

⁢

(

k

,

δ

)

are

e

c

⁢

(

log

⁡

1

δ

)

k

-

1

≤

N

⁢

(

k

,

δ

)

≤

2

2

δ

-

2

2

k

+

9

,

where the

lower bound

is due to Behrend

[

1

]

(for

k

=

3

) and Rankin

[

3

]

, and the

upper bound

is due to Gowers

[

2

]

.

For

k

=

3

a better upper bound was obtained by Bourgain

N

⁢

(

3

,

δ

)

≤

c

⁢

δ

-

2

⁢

e

2

56

⁢

δ

-

2

.

References

1

Felix A. Behrend.

On the sets of integers which contain no three in arithmetic
progression.

Proc. Nat. Acad. Sci.

, 23:331–332, 1946.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0060.10302

Zbl 0060.10302.

2

Timothy Gowers

.

A new proof of Szemerédi’s

theorem

.

Geom. Funct. Anal.

, 11(3):465–588, 2001.

Preprint available at

http://www.dpmms.cam.ac.uk/ wtg10/papers.html

http://www.dpmms.cam.ac.uk/ wtg10/papers.html.

3

Robert A. Rankin.

Sets of integers containing not more than a given number of terms in

arithmetical

progression.

Proc. Roy. Soc. Edinburgh Sect. A

, 65:332–344, 1962.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0104.03705

Zbl 0104.03705.

4

Klaus Friedrich Roth.

On certain sets of integers.

J. London Math. Soc.

, 28:245–252, 1953.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0050.04002

Zbl 0050.04002.

5

Endre Szemerédi.

On sets of integers containing no four

elements

in arithmetic
progression.

Acta Math. Acad. Sci. Hung.

, 20:89–104, 1969.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0175.04301

Zbl 0175.04301.

6

Endre Szemerédi.

On sets of integers containing no

k

elements in arithmetic
progression.

Acta. Arith.

, 27:299–345, 1975.

http://www.emis.de/cgi-bin/zmen/ZMATH/en/quick.html?type=html&an=0303.10056

Zbl 0303.10056.

Title

Szemerédi’s theorem

Canonical

name

SzemeredisTheorem

Date of creation

2013-03-22 13:19:40

Last modified on

2013-03-22 13:19:40

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

17

Author

bbukh (348)

Entry type

Theorem

Classification

msc 11B25

Classification

msc 05D10

Generated on Fri Feb 9 15:26:42 2018 by

LaTeXML
table of addition up to 12

table of addition up to 12

Because of the commutative property of

addition

, it does not matter if the row or the column gives the first operand.

+

1

2

3

4

5

6

7

8

9

10

11

12

1

2

3

4

5

6

7

8

9

10

11

12

13

2

3

4

5

6

7

8

9

10

11

12

13

14

3

4

5

6

7

8

9

10

11

12

13

14

15

4

5

6

7

8

9

10

11

12

13

14

15

16

5

6

7

8

9

10

11

12

13

14

15

16

17

6

7

8

9

10

11

12

13

14

15

16

17

18

7

8

9

10

11

12

13

14

15

16

17

18

19

8

9

10

11

12

13

14

15

16

17

18

19

20

9

10

11

12

13

14

15

16

17

18

19

20

21

10

11

12

13

14

15

16

17

18

19

20

21

22

11

12

13

14

15

16

17

18

19

20

21

22

23

12

13

14

15

16

17

18

19

20

21

22

23

24

It is

obvious

that the longest northwest to southeast diagonal contains numbers of the form

2

⁢

n

.

Title

table of addition up to 12

Canonical

name

TableOfAdditionUpTo12

Date of creation

2013-03-22 16:36:05

Last modified on

2013-03-22 16:36:05

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11B25

Classification

msc 00A05

Classification

msc 00A06

Generated on Fri Feb 9 15:26:43 2018 by

LaTeXML
table of subtraction up to 12

table of subtraction up to 12

In this table of

subtraction

, the column operand is first and the row operand is second.

-

1

2

3

4

5

6

7

8

9

10

11

12

1

0

-

1

-

2

-

3

-

4

-

5

-

6

-

7

-

8

-

9

-

10

-

11

2

1

0

-

1

-

2

-

3

-

4

-

5

-

6

-

7

-

8

-

9

-

10

3

2

1

0

-

1

-

2

-

3

-

4

-

5

-

6

-

7

-

8

-

9

4

3

2

1

0

-

1

-

2

-

3

-

4

-

5

-

6

-

7

-

8

5

4

3

2

1

0

-

1

-

2

-

3

-

4

-

5

-

6

-

7

6

5

4

3

2

1

0

-

1

-

2

-

3

-

4

-

5

-

6

7

6

5

4

3

2

1

0

-

1

-

2

-

3

-

4

-

5

8

7

6

5

4

3

2

1

0

-

1

-

2

-

3

-

4

9

8

7

6

5

4

3

2

1

0

-

1

-

2

-

3

10

9

8

7

6

5

4

3

2

1

0

-

1

-

2

11

10

9

8

7

6

5

4

3

2

1

0

-

1

12

11

10

9

8

7

6

5

4

3

2

1

0

The longest northwest to southeast diagonal obviously contains zeroes.

Title

table of subtraction up to 12

Canonical

name

TableOfSubtractionUpTo12

Date of creation

2013-03-22 16:36:11

Last modified on

2013-03-22 16:36:11

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11B25

Classification

msc 00A05

Classification

msc 00A06

Generated on Fri Feb 9 15:26:45 2018 by

LaTeXML
amenable number

amenable number

An

amenable number

is an

integer

for which there exists a multiset of at least two

positive integers

that can be either added up or multiplied together to give the original number. To put it algebraically, for an integer

n

, there is a set of integers

a

1

,

…

,

a

k

, for which the equalities

n

=

∑

i

=

1

k

a

i

=

∏

i

=

1

k

a

i

hold. An elegant example is the number 6, which is an amenable number since

6

=

1

+

2

+

3

=

1

⋅

2

⋅

3

.

All

composite numbers

are amenable because, even if other

solutions

are available, one can always make an inelegant solution by taking the

prime factorization

(expressed with repeated factors rather than

exponents

) and add as many 1s as necessary to add up to

n

. Because of the

multiplicative identity

, multiplying this set of integers will yield

n

no matter how many 1s there are in the set.

If we allow negative integers in the set, then all

prime numbers

are also amenable. Even if no other solutions are available, one can always make an inelegant solution for a prime number

p

of

1

,

-

1

,

1

,

-

1

,

p

. In the sum, the two

positive

ones are cancelled out by the two negative ones, leaving

p

, while in the product, the two negative ones cancel out the effect of their signs, leaving only the multiplicative identity to affect the final result.

Amenable numbers should not be confused with

amicable numbers

, which are pairs of integers whose

divisors

add up to each other.

Title

amenable number

Canonical name

AmenableNumber

Date of creation

2013-03-22 16:37:02

Last modified on

2013-03-22 16:37:02

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11B34

Generated on Fri Feb 9 15:26:47 2018 by

LaTeXML