aarjaneiro/planetmath huggingface dataset:
non-multiplicative function

non-multiplicative function

In

number theory

, a

non-multiplicative function

is an

arithmetic function

which is not

multiplicative

.

Examples

Some examples of a non-multiplicative functions are the arithmetic functions:

•

r

2

⁢

(

n

)

- the number of unordered representations of

n

as a sum of squares of two

integers

,

positive

, negative or zero,

•

c

4

⁢

(

n

)

- the number of ways that

n

can be expressed as the sum of four squares of

nonnegative integers

, where we distinguish between different orders of the summands. For example:

1

=

1

2

+

0

2

+

0

2

+

0

2

=

0

2

+

1

2

+

0

2

+

0

2

+

0

2

=

0

2

+

0

2

+

1

2

+

0

2

=

0

2

+

0

2

+

0

2

+

1

2

,

hence

c

4

⁢

(

1

)

=

4

≠

1

.

•

The

partition function

P

⁢

(

n

)

- the number of ordered representations of

n

as a sum of

positive integers

. For instance:

P

⁢

(

2

⋅

5

)

=

P

⁢

(

10

)

=

42

and

P

⁢

(

2

)

⁢

P

⁢

(

5

)

=

2

⋅

7

=

14

≠

42

.

•

The

prime counting function

π

⁢

(

n

)

. Here we first have

π

⁢

(

1

)

=

0

≠

1

and then we have as yet for example:

π

⁢

(

2

⋅

5

)

=

π

⁢

(

10

)

=

4

and

π

⁢

(

2

)

⁢

π

⁢

(

5

)

=

1

⋅

3

=

3

≠

4

.

•

The

Mangoldt function

Λ

⁢

(

n

)

.

Λ

⁢

(

1

)

=

ln

⁡

1

≠

1

and for example:

Λ

⁢

(

2

⋅

5

)

=

Λ

⁢

(

10

)

=

0

and

Λ

⁢

(

2

)

⁢

Λ

⁢

(

5

)

=

ln

⁡

2

⋅

ln

⁡

5

≠

0

.

We would think that for some

n

multiplicativity of

Λ

⁢

(

n

)

would be true as in:

Λ

⁢

(

2

⋅

6

)

=

Λ

⁢

(

12

)

=

0

and

Λ

⁢

(

2

)

⁢

Λ

⁢

(

6

)

=

ln

⁡

2

⋅

0

=

0

,

but we have to write:

Λ

⁢

(

2

2

)

⁢

Λ

⁢

(

3

)

=

ln

⁡

2

⋅

ln

⁡

3

≠

0

.

Title

non-multiplicative function

Canonical

name

NonmultiplicativeFunction

Date of creation

2013-03-22 12:47:04

Last modified on

2013-03-22 12:47:04

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

16

Author

Mathprof (13753)

Entry type

Example

Classification

msc 11A25

Related topic

PartitionFunction2

Defines

partition function

Generated on Fri Feb 9 15:13:59 2018 by

LaTeXML
nonprincipal real characters @⁢ERROR(\symoperators)⁢m⁢o⁢d⁢ERROR(\tmspace)+.1667⁢e⁢m⁢ERROR(\tmspace)+.1667⁢e⁢m⁢p are unique

nonprincipal real characters

@

⁢

\symoperators

⁢

m

⁢

o

⁢

d

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

p

are unique

Theorem 1

Let

p

be a prime. Then there is a unique nonprincipal real

Dirichlet character

χ

mod

p

, given by

χ

⁢

(

n

)

=

(

n

p

)

Proof.

Note first that

χ

⁢

(

n

)

=

(

n

p

)

is obviously a nonprincipal real

character

mod

p

. Now, suppose

χ

is any nonprincipal real character

mod

p

. Choose some

generator

,

a

, of

(

ℤ

/

p

⁢

ℤ

)

*

. Clearly

χ

⁢

(

a

)

=

-

1

(since otherwise

χ

is principal), and thus

χ

⁢

(

a

k

)

=

(

-

1

)

k

. Since

a

p

-

1

=

1

, and no lower power of

a

is

1

, it follows that

χ

is

-

1

on exactly

(

p

-

1

)

/

2

elements of

(

ℤ

/

p

⁢

ℤ

)

*

and is

1

on exactly

(

p

-

1

)

/

2

elements. However, since

χ

⁢

(

x

2

)

=

χ

⁢

(

x

)

2

=

1

,

χ

is

1

on each of the

(

p

-

1

)

/

2

squares in

(

ℤ

/

p

⁢

ℤ

)

*

. Thus

χ

is

1

on squares and

-

1

on nonsquares, so

χ

⁢

(

n

)

=

(

n

p

)

.

Note that this result is not true if

p

is not prime. For example, the following is a table of Dirichlet characters modulo

8

, all of which are real:

χ

1

χ

2

χ

3

χ

4

1

1

1

1

1

3

1

1

-

1

-

1

5

1

-

1

1

-

1

7

1

-

1

-

1

1

Title

nonprincipal real characters

@

⁢

\symoperators

⁢

m

⁢

o

⁢

d

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

\tmspace

+

.1667

⁢

e

⁢

m

⁢

p

are unique

Canonical name

NonprincipalRealCharactersmodPAreUnique

Date of creation

2013-03-22 16:34:51

Last modified on

2013-03-22 16:34:51

Owner

rm50 (10146)

Last modified by

rm50 (10146)

Numerical id

6

Author

rm50 (10146)

Entry type

Theorem

Classification

msc 11A25

Generated on Fri Feb 9 15:14:01 2018 by

LaTeXML
nontotient

nontotient

An

integer

n

>

0

is called a

nontotient

if

N

ϕ

⁢

(

n

)

=

0

, where

N

ϕ

is the

totient valence function

. This is the case of any

odd integer

greater than 1. This is also the case of twice a prime that isn’t a

Sophie Germain prime

, one more than a prime, an

oblong number

that is the product of a prime and one more or less than that prime, etc.

Sloane’s OEIS lists even values in A005277 and all values together in A007617.

Title

nontotient

Canonical name

Nontotient

Date of creation

2013-03-22 15:51:20

Last modified on

2013-03-22 15:51:20

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Related topic

Noncototient

Generated on Fri Feb 9 15:14:02 2018 by

LaTeXML
number of distinct prime factors function

number of distinct prime factors function

The

number of distinct prime factors function

ω

⁢

(

n

)

counts how many

distinct prime factors

n

has. Expressing

n

as

n

=

∏

i

=

1

k

p

i

a

i

,

where the

p

i

are distinct primes, there being

k

of them, and the

a

i

are

positive integers

(not necessarily distinct), then

ω

⁢

(

n

)

=

k

.

Obviously for a prime

p

it follows that

ω

⁢

(

p

)

=

1

. When

n

is a

squarefree

number, then

Ω

⁢

(

n

)

=

ω

⁢

(

n

)

, where

Ω

⁢

(

n

)

is the number of (nondistinct)

prime factors

function

(

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

). Otherwise,

Ω

⁢

(

n

)

>

ω

⁢

(

n

)

.

ω

⁢

(

n

)

is an

additive function

, and it can be used to define a

multiplicative function

like the Möbius function

μ

⁢

(

n

)

=

(

-

1

)

ω

⁢

(

n

)

(as long as

n

is squarefree).

Title

number of distinct prime factors function

Canonical name

NumberOfDistinctPrimeFactorsFunction

Date of creation

2013-03-22 16:06:52

Last modified on

2013-03-22 16:06:52

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

9

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A25

Related topic

NumberOfNondistinctPrimeFactorsFunction

Related topic

2omeganLeTaunLe2Omegan

Generated on Fri Feb 9 15:14:04 2018 by

LaTeXML
number of (nondistinct) prime factors function

number of (nondistinct) prime factors function

The

Ω

⁢

(

n

)

counts with repetition how many

prime factors

a

natural number

n

has. If

n

=

∏

j

=

1

k

p

j

a

j

where the

k

primes

p

j

are distinct and the

a

j

are natural numbers, then

Ω

⁢

(

n

)

=

∑

j

=

1

k

a

j

.

Note that, if

n

is a

squarefree

number, then

ω

⁢

(

n

)

=

Ω

⁢

(

n

)

, where

ω

⁢

(

n

)

is the

number of distinct prime factors function

. Otherwise,

ω

⁢

(

n

)

<

Ω

⁢

(

n

)

.

Note also that

Ω

⁢

(

n

)

is a

completely additive function

and thus can be exponentiated to define a

completely multiplicative function

. For example, the

Liouville function

can be defined as

λ

⁢

(

n

)

=

(

-

1

)

Ω

⁢

(

n

)

.

The

sequence

{

Ω

⁢

(

n

)

}

appears in the OEIS as sequence

http://www.research.att.com/ njas/sequences/?q=A001222

A001222.

The sequence

{

2

Ω

⁢

(

n

)

}

appears in the OEIS (

http://planetmath.org/OEIS

) as sequence

http://www.research.att.com/ njas/sequences/?q=A061142

A061142.

Title

number of (nondistinct) prime factors

function

Canonical name

NumberOfnondistinctPrimeFactorsFunction

Date of creation

2013-03-22 16:07:00

Last modified on

2013-03-22 16:07:00

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Definition

Classification

msc 11A25

Related topic

NumberOfDistinctPrimeFactorsFunction

Related topic

2omeganLeTaunLe2Omegan

Generated on Fri Feb 9 15:14:05 2018 by

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parity of τ function

parity of

τ

function

If the

prime factor

decomposition of a

positive integer

n

is

n

=

p

1

α

1

⁢

p

2

α

2

⁢

⋯

⁢

p

r

α

r

,

(1)

then all

positive

divisors

of

n

are of the form

p

1

ν

1

p

2

ν

2

⋯

p

r

ν

r

where

0

≤

ν

i

≤

α

i

(

i

=

1

,

2

,

…

,

r

)

.

Thus the total number of the divisors is

τ

⁢

(

n

)

=

(

α

1

+

1

)

⁢

(

α

2

+

1

)

⁢

⋯

⁢

(

α

r

+

1

)

.

(2)

From this we see that in to

τ

⁢

(

n

)

be an

odd number

, every sum

α

i

+

1

shall be odd, i.e. every

exponent

α

i

in (1) must be even.  It means that

n

has an

even number

of each of its

prime divisors

p

i

; so

n

is a square of an

integer

, a

perfect square

.

Consequently, the number of all positive divisors of an integer is always even, except if the integer is a perfect square.

Examples

.  15 has four positive divisors 1, 3, 5, 15 and the

square number

16 five divisors

1, 2, 4, 8, 16.

Title

parity of

τ

function

Canonical name

ParityOftauFunction

Date of creation

2013-03-22 18:55:43

Last modified on

2013-03-22 18:55:43

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

6

Author

pahio (2872)

Entry type

Feature

Classification

msc 11A25

Related topic

TauFunction

Generated on Fri Feb 9 15:14:07 2018 by

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perfect totient number

perfect totient number

An

integer

n

is a

perfect totient number

if

n

=

∑

i

=

1

c

+

1

ϕ

i

⁢

(

n

)

, where

ϕ

i

⁢

(

x

)

is the

iterated totient function

and

c

is the integer such that

ϕ

c

⁢

(

n

)

=

2

.

A082897 in Sloane’s OEIS lists the first few perfect totient numbers: 3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, etc. It can be observed that many of these are

multiples

of 3 (in fact, 4375 is the smallest one that is not

divisible

by 3) and in fact all

3

x

for

x

>

0

are perfect totient numbers.

Furthermore,

3

⁢

p

for a prime

p

>

3

is a perfect totient number if and only if

p

=

4

⁢

n

+

1

, where

n

itself is also a perfect totient number. Mohan and Suryanarayana showed why

3

⁢

p

can’t be a perfect totient number when

p

≡

3

mod

4

. In regards to

3

2

⁢

p

, Ianucci et al showed that if it is a perfect totient number then

p

is a prime of one of three specific forms listen in their paper. It is not known if there are any perfect totient numbers of the form

3

x

⁢

p

for

x

>

3

.

References

1

Perez Cacho, “On the sum of

totients

of successive orders,”

Revista Matematica Hispano-Americana

5.3

(1939): 45 - 50

2

D. E. Ianucci, D. Moujie & G. L. Cohen, “On Perfect Totient Numbers”

Journal of

Integer Sequences

,

6

, 2003: 03.4.5

3

R. K. Guy,

Unsolved Problems in

Number Theory

New York: Springer-Verlag 2004: B42

Title

perfect totient number

Canonical

name

PerfectTotientNumber

Date of creation

2013-03-22 16:33:20

Last modified on

2013-03-22 16:33:20

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A25

Synonym

totient perfect number

Related topic

IteratedTotientFunction

Generated on Fri Feb 9 15:14:09 2018 by

LaTeXML
pointwise multiplication of a completely multiplicative function distibutes over convolution

pointwise multiplication of a completely multiplicative function distibutes over convolution

Theorem

.

Let

f

be a

completely multiplicative function

and

g

and

h

be

arithmetic functions

. Then

f

⁢

(

g

*

h

)

=

(

f

⁢

g

)

*

(

f

⁢

h

)

.

Proof.

Let

n

be a

positive integer

. Then

(

f

⁢

(

g

*

h

)

)

⁢

(

n

)

=

f

⁢

(

n

)

⁢

(

g

*

h

)

⁢

(

n

)

=

f

⁢

(

n

)

⁢

∑

d

|

n

g

⁢

(

d

)

⁢

h

⁢

(

n

d

)

=

∑

d

|

n

f

⁢

(

n

)

⁢

g

⁢

(

d

)

⁢

h

⁢

(

n

d

)

=

∑

d

|

n

f

⁢

(

d

⋅

n

d

)

⁢

g

⁢

(

d

)

⁢

h

⁢

(

n

d

)

=

∑

d

|

n

f

⁢

(

d

)

⁢

f

⁢

(

n

d

)

⁢

g

⁢

(

d

)

⁢

h

⁢

(

n

d

)

=

∑

d

|

n

(

f

⁢

g

)

⁢

(

d

)

⁢

(

f

⁢

h

)

⁢

(

n

d

)

=

(

(

f

⁢

g

)

*

(

f

⁢

h

)

)

⁢

(

n

)

.

∎

Title

pointwise

multiplication

of a completely multiplicative function distibutes over convolution

Canonical name

PointwiseMultiplicationOfACompletelyMultiplicativeFunctionDistibutesOverConvolution

Date of creation

2013-03-22 15:59:49

Last modified on

2013-03-22 15:59:49

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

8

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 11A25

Related topic

ArithmeticFunction

Related topic

CompletelyMultiplicative

Related topic

MultiplicativeFunction

Generated on Fri Feb 9 15:14:11 2018 by

LaTeXML
polite number

polite number

A

polite number

n

is an

integer

that is the sum of two or more consecutive

nonnegative integers

in at least one way. To put it algebraically, if

n

is polite then there is a solution to

n

=

∑

i

=

a

b

i

with

b

>

a

and

a

>

-

1

. For example, 42 is a polite number since it is the sum of the integers from 3 to 9. The first few polite numbers are 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, etc.

Obviously all

triangular numbers

are polite numbers. So are all

odd numbers

. In fact, the numbers that are not polite are the powers of 2.

Title

polite number

Canonical name

PoliteNumber

Date of creation

2013-03-22 18:09:54

Last modified on

2013-03-22 18:09:54

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:14:13 2018 by

LaTeXML
Pólya’s conjecture

Pólya’s conjecture

(George Pólya) Given any range of consecutive

integers

from 1 to

n

>

1

, at least half, if not more, of the integers in that range will have an

odd number

of

prime factors

(not necessarily distinct). Or, restated using the

Liouville function

λ

⁢

(

i

)

=

(

-

1

)

Ω

⁢

(

i

)

(where

Ω

⁢

(

i

)

is the number of (nondistinct) prime factors

function

(

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

)), there is no such

n

>

1

such that

L

⁢

(

n

)

>

0

where

L

⁢

(

n

)

=

∑

i

=

1

n

λ

⁢

(

i

)

.

The zeroes of the sum of the Liouville function below 1000 (namely 2, 4, 6, 10, 16, 26, 40, 96, 586, listed in A028488 of Sloane’s OEIS) were known early on after the

conjecture

was posed. These are all followed by primes, with the exception of 26, which is followed by

3

3

.

Arthur Ingham proved the conjecture false in 1942 and gave a method for finding

counterexamples

, but the first counterexample wasn’t found until 1960 by Robert Lehman, namely

n

=

906180359

. Two decades later, Minoru Tanaka found the smallest counterexample at

n

=

906150257

; the next three integers have 2, 4 or 6 non-distinct prime factors.

Title

Pólya’s conjecture

Canonical

name

PolyasConjecture

Date of creation

2013-03-22 16:39:43

Last modified on

2013-03-22 16:39:43

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Conjecture

Classification

msc 11A25

Synonym

Polya’s conjecture

Synonym

Polya conjecture

Synonym

Pólya conjecture

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

LaTeXML
power of two

power of two

A

power of two

is a number of the form

2

n

, with

n

generally understood to be a

nonnegative integer

. The first few powers of two are 1, 2, 4, 8, 16, 32, 64, 128 and so forth. These are listed in A000079 of Sloane’s OEIS. Because

computers

use the binary numeral system for their computations, the powers of two are of paramount importance in computer science.

With

n

a negative integer, the larger powers of two are the

fractions

1

2

,

1

4

,

1

8

,

1

16

,

1

32

,

1

64

,

1

128

,

1

256

, which are so often used in

conjunction

with units of the British Weights and

Measures

Act of which some have been inherited and are still used in the United States. These fractions are also used in music for gradually smaller rhythmic units, as well as in photography, though somewhat relabelled (e.g., the readout of a camera might read

1

250

rather than

1

256

).

The binary

representation

of an power of two greater than 1 is always a digit 1 as the most significant digit followed by as many zeroes as indicated by the

exponent

for 2, such as

2

4

is in binary

10000

2

. Hence

2

n

⁢

XOR

⁢

2

n

-

1

=

0

.

For

2

n

, the value of

τ

⁢

(

2

n

)

(the

divisor function

) is always

n

+

1

, and the

divisors

are the powers of two from

2

0

to

2

n

. The value of the

sum of divisors function

σ

⁢

(

2

n

)

is therefore a

Mersenne number

2

n

+

1

-

1

. The powers of two are therefore

deficient numbers

always 1 short of being a

perfect numbers

. In fact, they are the only known almost perfect numbers. So the sum of consecutive powers of two from

i

=

-

1

down to some other

negative

i

is a fraction with a Mersenne number as its numerator and a power of two as its denominator. This suggests that

lim

n

→

∞

⁡

∑

i

=

1

n

1

2

i

=

1

.

Only a

Collatz sequence

starting with a power of two is in

strictly descending order

, and will reach 1 in precisely

n

steps. Any other starting value guarantees that there will be steps at which the value will be tripled and incremented rather than just halved.

For

n

>

0

, a number

2

n

is an

impolite number

, it cannot be represented as the sum of

consecutive

nonnegative integers, whereas all other

positive integers

can be (and are thus

polite numbers

).

Title

power of two

Canonical

name

PowerOfTwo

Date of creation

2013-03-22 18:09:56

Last modified on

2013-03-22 18:09:56

Owner

1and2and4 (20899)

Last modified by

1and2and4 (20899)

Numerical id

4

Author

1and2and4 (20899)

Entry type

Definition

Classification

msc 11A25

Related topic

AllPositiveIntegersArePoliteNumbersExceptPowersOfTwo

Defines

impolite number

Generated on Fri Feb 9 15:14:16 2018 by

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practical number

practical number

A

positive integer

m

is called a

practical number

if every positive integer

n

<

m

is a sum of distinct

positive

divisors

of

m

.

.

An

integer

m

≥

2

,

m

=

p

1

α

1

⁢

p

2

α

2

⁢

⋯

⁢

p

ℓ

α

ℓ

,

with primes

p

1

<

p

2

<

…

<

p

ℓ

and integers

α

i

≥

1

,

is practical if and only if

p

1

=

2

and, for

i

=

2

,

3

,

…

,

ℓ

,

p

i

≤

σ

(

p

1

α

1

p

2

α

2

⋯

p

i

-

1

)

α

i

-

1

+

1

,

where

σ

⁢

(

n

)

denotes the sum of the positive divisors of

n

.

Let

P

⁢

(

x

)

be the counting function of practical numbers.
Saias

[

2

]

, using suitable sieve methods introduced by Tenenbaum

[

3

,

4

]

, proved
a good estimate in terms of
a Chebishev-type

theorem

: for suitable
constants

c

1

and

c

2

,

c

1

⁢

x

log

⁡

x

<

P

⁢

(

x

)

<

c

2

⁢

x

log

⁡

x

.

In

[

1

]

Melfi proved a Goldbach-type result showing
that every even positive integer is a sum of two practical numbers, and that there exist infinitely many triplets of practical numbers of the form

m

-

2

,

m

,

m

+

2

.

References

1

G. Melfi, On two

conjectures

about practical numbers,

J.

Number Theory

56

(1996), 205–210.

2

E. Saias, Entiers à
diviseurs denses 1,

J. Number Theory

62

(1997), 163–191.

3

G. Tenenbaum, Sur un problème de crible et ses applications,
Ann. Sci.
Éc. Norm. Sup. (4)

19

(1986), 1–30.

4

G. Tenenbaum, Sur un problème de crible et ses
applications, 2.
Corrigendum et étude du graphe divisoriel, Ann. Sci. Éc. Norm. Sup.
(4)

28

(1995), 115–127.

Title

practical number

Canonical

name

PracticalNumber

Date of creation

2013-03-22 14:12:17

Last modified on

2013-03-22 14:12:17

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

6

Author

mathcam (2727)

Entry type

Definition

Classification

msc 11A25

Generated on Fri Feb 9 15:14:18 2018 by

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practical number

practical number

The second definition of a

practical number

is: a

positive integer

n

such that each smaller

integer

m

can be represented as a sum of distinct

proper divisors

d

i

of

n

(for

1

<

i

<

τ

⁢

(

n

)

, with

τ

⁢

(

n

)

being the

divisor function

; 1 is not considered a proper divisor for this application). The first few are 6, 12, 18, 20, 24, 28, 30, 36, 40, 42, 48, 54, 56, 60, 66, 72, 78, 80, 84, 88, 90, 96, 100, etc., listed in A007620 of Sloane’s OEIS.

For example, 12 is practical. It has for

divisors

1, 2, 3, 4, 6 and 12, but only 2, 3, 4 and 6 are considered proper divisors here. The sum can consist of a single summand, so we need only concern ourselves with numbers less than 12 that are not divisors of 12. We verify that indeed 2 + 3 = 5, 3 + 4 = 7, 2 + 6 = 8, 3 + 6 = 9, 4 + 6 = 10 and 2 + 3 + 6 = 11.

Under this definition, the powers of 2 are not practical numbers. Representing

odd numbers

smaller than a power of 2 requires using 1 in the sums of divisors.

Title

practical number

Canonical name

PracticalNumber1

Date of creation

2013-03-22 18:07:03

Last modified on

2013-03-22 18:07:03

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A25

Related topic

ImpracticalNumber

Generated on Fri Feb 9 15:14:20 2018 by

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prime counting function

prime counting function

The

prime counting function

is a

non-multiplicative function

for any positive real number

x

, denoted as

π

⁢

(

x

)

and gives the number of primes not exceeding

x

. It usually takes a

positive integer

n

for an argument. The first few values of

π

⁢

(

n

)

for

n

=

1

,

2

,

3

,

…

are

0

,

1

,

2

,

2

,

3

,

3

,

4

,

4

,

4

,

4

,

5

,

5

,

6

,

6

,

6

,

6

,

7

,

7

,

8

,

8

⁢

…

(

http://www.research.att.com/ njas/

sequences

/eisA.cgi?Anum=000720

OEIS A000720
).

The asymptotic behavior of

π

⁢

(

x

)

∼

x

/

ln

⁡

x

is given by the

prime number theorem

. This function is closely related with

Chebyshev’s functions

ϑ

⁢

(

x

)

and

ψ

⁢

(

x

)

.

Title

prime counting function

Canonical name

PrimeCountingFunction

Date of creation

2013-03-22 12:49:00

Last modified on

2013-03-22 12:49:00

Owner

XJamRastafire (349)

Last modified by

XJamRastafire (349)

Numerical id

13

Author

XJamRastafire (349)

Entry type

Definition

Classification

msc 11A25

Classification

msc 11A41

Classification

msc 11N05

Related topic

LogarithmicIntegral2

Generated on Fri Feb 9 15:14:21 2018 by

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product of divisors function

product of divisors function

The product of all

positive

divisors

of a nonzero

integer

n

is equal

n

τ

⁢

(

n

)

, where tau function

τ

⁢

(

n

)

expresses the number of the positive divisors of

n

.

Proof.

Let

t

=

τ

⁢

(

n

)

and the positive divisors of

n

be

a

1

<

a

2

<

…

<

a

t

.

If

n

is not a square of an integer,

t

is even (see

http://planetmath.org/node/11781

parity of

τ

function), whence

{

a

1

⁢

a

t

=

n

a

2

⁢

a

t

-

1

=

n

⋯

a

t

2

⁢

a

t

+

2

2

=

n

.

Thus

∏

d

∣

n

d

=

a

1

⁢

a

2

⁢

⋯

⁢

a

t

=

n

t

2

.

If

n

is a square of an integer,

t

is odd, and we have

{

a

1

⁢

a

t

=

n

a

2

⁢

a

t

-

1

=

n

⋯

a

t

-

1

2

⁢

a

t

+

3

2

=

n

a

t

+

1

2

=

n

1

2

.

In this case we obtain a result:

∏

d

∣

n

d

=

a

1

⁢

a

2

⁢

⋯

⁢

a

t

=

n

t

-

1

2

+

1

2

=

n

t

2

Note.

The

absolute value

of the product of all divisors is

n

τ

⁢

(

n

)

.

Title

product of divisors function

Canonical name

ProductOfDivisorsFunction

Date of creation

2013-03-22 18:55:45

Last modified on

2013-03-22 18:55:45

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Definition

Classification

msc 11A25

Synonym

divisor product

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