aarjaneiro/planetmath huggingface dataset:
multiply perfect number

multiply perfect number

A

multiply perfect number

n

for a given

k

is an

integer

such that

σ

⁢

(

n

)

=

k

⁢

n

, where

σ

⁢

(

x

)

is the

sum of divisors function

.

n

is then called

k

-

perfect

. For example, 120 is 3-perfect since its

divisors

(1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120) add up to 360, which is thrice 120. Numbers that are 2-perfect are by default called

perfect numbers

.

The 3-perfect numbers are listed in A005820 of Sloane’s OEIS; A027687 lists 4-perfect numbers; etc. The first

k

-perfect number for

k

>

1

are listed in A007539.

As of 2007, more than 5000 multiply perfect numbers were known. The Multiply Perfect Numbers webpage, hosted by Bielefeld University, provides an ASCII text file database giving the name of the discoverer, “abundancy” (the value of

k

for the particular

n

), some of the

prime factors

and the value of

log

⁡

log

⁡

n

.

References

1

A. H. Beiler,

Recreations in the Theory of Numbers

. New York (1964): 22.

2

A. L. Brown, “Multiperfect numbers”

Scripta Math.

20

(1954): 103 - 106

Title

multiply perfect number

Canonical name

MultiplyPerfectNumber

Date of creation

2013-03-22 17:47:58

Last modified on

2013-03-22 17:47:58

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Synonym

multiperfect number

Synonym

pluperfect number

Generated on Fri Feb 9 15:10:16 2018 by

LaTeXML
Ore number

Ore number

Given a

positive integer

n

with

divisors

d

1

,

…

,

d

k

,

if the

harmonic mean

k

∑

i

=

1

k

1

d

i

∈

ℤ

,

then

n

is an

Ore number

or

harmonic divisor number

.

For example, 270 has the divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, 135 and 270. The

reciprocals

of these 16 divisors add up to

8

3

. Then 16 divided by that

fraction

is 6, an

integer

. Thus 270 is an Ore number.

The first few Ore numbers are 1, 6, 28, 140, 270, 496, 672, 1638, 2970, listed in A001599 of Sloane’s OEIS.

All even perfect numbers are Ore numbers, a fact proven by Øystein Ore in 1948.

1 is the only known odd Ore number. If there’s another, it would have to be pretty big, and is considered as unlikely to exist as an odd perfect number.

Title

Ore number

Canonical

name

OreNumber

Date of creation

2013-03-22 15:56:28

Last modified on

2013-03-22 15:56:28

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

15

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Synonym

harmonic divisor number

Generated on Fri Feb 9 15:10:18 2018 by

LaTeXML
positive multiple of a semiperfect number is also semiperfect

positive multiple of a semiperfect number is also semiperfect

Just as the

theorem on multiples of abundant numbers

shows that

multiples

of

abundant numbers

are also abundant, it is also true that multiples of

semiperfect numbers

are also semiperfect, and T. Foregger’s proof of the abundant number theorem lays bare a simple mechanism that we can also employ for semiperfect numbers.

Given the

divisors

d

1

,

…

,

d

k

-

1

of

n

(where

k

=

τ

⁢

(

n

)

and

τ

⁢

(

x

)

is the

divisor function

), sorted in

ascending order

for our convenience, and with a smart iterator

i

that somehow knows to skip over those divisors that contribute to

n

’s

abundance

, we can show that the divisors of

n

⁢

m

(with

m

>

0

) will include

d

1

⁢

m

,

…

,

d

k

-

1

⁢

m

. With our smart iterator

i

and thanks to the

distributive property

of

multiplication

, it follows that

∑

i

=

1

k

-

1

d

i

⁢

m

=

n

⁢

m

,

our desired result.

Title

positive

multiple of a semiperfect number is also semiperfect

Canonical name

PositiveMultipleOfASemiperfectNumberIsAlsoSemiperfect

Date of creation

2013-03-22 16:18:57

Last modified on

2013-03-22 16:18:57

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Derivation

Classification

msc 11A05

Generated on Fri Feb 9 15:10:22 2018 by

LaTeXML
positive multiple of an abundant number is abundant

positive multiple of an abundant number is abundant

Theorem.

A

positive

multiple

of an

abundant number

is abundant.

Proof.

Let

n

be abundant and

m

>

0

be an

integer

. We have to show
that

σ

⁢

(

m

⁢

n

)

>

2

⁢

m

⁢

n

, where

σ

⁢

(

n

)

is the sum of the positive

divisors

of

n

.
Let

d

1

,

…

,

d

k

be the positive divisors of

n

. Then
certainly

m

⁢

d

1

,

…

,

m

⁢

d

k

are distinct divisors of

m

⁢

n

. The result is
clear if

m

=

1

, so assume

m

>

1

. Then

σ

⁢

(

m

⁢

n

)

>

1

+

∑

i

=

1

k

m

⁢

d

i

>

m

⁢

∑

i

=

1

k

d

i

>

m

⁢

(

2

⁢

n

)

=

2

⁢

m

⁢

n

.

As a corollary, the positive abundant numbers form a

semigroup

.

Title

positive multiple of an abundant number is abundant

Canonical name

PositiveMultipleOfAnAbundantNumberIsAbundant

Date of creation

2013-03-22 16:17:07

Last modified on

2013-03-22 16:17:07

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

13

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 11A05

Related topic

TheoremOnMultiplesOfAbundantNumbers

Generated on Fri Feb 9 15:10:24 2018 by

LaTeXML
proof of Chinese remainder theorem

proof of Chinese remainder theorem

First we prove that

𝔞

i

+

∏

j

≠

i

𝔞

j

=

R

for each

i

. Without loss of generality, assume that

i

=

1

. Then

R

=

(

𝔞

1

+

𝔞

2

)

⁢

(

𝔞

1

+

𝔞

3

)

⁢

⋯

⁢

(

𝔞

1

+

𝔞

n

)

,

since each factor

𝔞

1

+

𝔞

j

is

R

. Expanding the

product

, each term will contain

𝔞

1

as a factor, except the term

𝔞

2

⁢

𝔞

2

⁢

⋯

⁢

𝔞

n

. So we have

(

𝔞

1

+

𝔞

2

)

⁢

(

𝔞

1

+

𝔞

3

)

⁢

⋯

⁢

(

𝔞

1

+

𝔞

n

)

⊆

𝔞

1

+

𝔞

2

⁢

𝔞

2

⁢

⋯

⁢

𝔞

n

,

and hence the expression on the

right hand side

must equal

R

.

Now we can prove that

∏

𝔞

i

=

⋂

𝔞

i

, by

induction

. The statement is

trivial

for

n

=

1

. For

n

=

2

, note that

𝔞

1

∩

𝔞

2

=

(

𝔞

1

∩

𝔞

2

)

⁢

R

=

(

𝔞

1

∩

𝔞

2

)

⁢

(

𝔞

1

+

𝔞

2

)

⊆

𝔞

2

⁢

𝔞

1

+

𝔞

1

⁢

𝔞

2

=

𝔞

1

⁢

𝔞

2

,

and the reverse

inclusion

is obvious, since each

𝔞

i

is an ideal.
Assume that the statement is proved for

n

-

1

, and condsider it for

n

. Then

⋂

1

n

𝔞

i

=

𝔞

1

∩

⋂

2

n

𝔞

i

=

𝔞

1

∩

∏

2

n

𝔞

i

,

using the

induction hypothesis

in the last step.
But using the fact proved above and the

n

=

2

case, we see that

𝔞

1

∩

∏

2

n

𝔞

i

=

𝔞

1

⋅

∏

2

n

𝔞

i

=

∏

1

n

𝔞

i

.

Finally, we are ready to prove the . Consider the

ring homomorphism

R

→

∏

R

/

𝔞

i

defined by

projection

on each

component

of the product:

x

↦

(

𝔞

1

+

x

,

𝔞

2

+

x

,

…

,

𝔞

n

+

x

)

. It is easy to see that the kernel of this map is

⋂

𝔞

i

, which is also

∏

𝔞

i

by the earlier part of the proof. So it only remains to show that the map is

surjective

.

Accordingly, take an arbitrary

element

(

𝔞

1

+

x

1

,

𝔞

2

+

x

2

,

…

,

𝔞

n

+

x

n

)

of

∏

R

/

𝔞

i

. Using the first part of the proof, for each

i

, we can find elements

y

i

∈

𝔞

i

and

z

i

∈

∏

j

≠

i

𝔞

j

such that

y

i

+

z

i

=

1

. Put

x

=

x

1

⁢

z

1

+

x

2

⁢

z

2

+

…

+

x

n

⁢

z

n

.

Then for each

i

,

𝔞

i

+

x

=

𝔞

i

+

x

i

⁢

z

i

,

since

x

j

⁢

z

j

∈

𝔞

i

for all

j

≠

i

,

=

𝔞

i

+

x

i

⁢

y

i

+

x

i

⁢

z

i

,

since

x

i

⁢

y

i

∈

𝔞

i

,

=

𝔞

i

+

x

i

⁢

(

y

i

+

z

i

)

=

𝔞

i

+

x

i

⋅

1

=

𝔞

i

+

x

i

.

Thus the map is surjective as required, and induces the

isomorphism

R

∏

𝔞

i

→

∏

R

𝔞

i

.

Title

proof of Chinese remainder theorem

Canonical name

ProofOfChineseRemainderTheorem

Date of creation

2013-03-22 12:57:20

Last modified on

2013-03-22 12:57:20

Owner

mclase (549)

Last modified by

mclase (549)

Numerical id

8

Author

mclase (549)

Entry type

Proof

Classification

msc 11A05

Classification

msc 11N99

Classification

msc 13A15

Generated on Fri Feb 9 15:10:27 2018 by

LaTeXML
proof of long division

proof of long division

Proof of theorem 1.

Let

a

,

b

be

integers

,

b

≠

0

. Set

q

=

{

⌊

a

b

⌋

if

b

>

0

-

⌊

a

|

b

|

⌋

otherwise

,

and

r

=

a

-

q

⋅

b

. Since

0

≤

x

-

⌊

x

⌋

<

1

for any real

x

, we get for

positive

b

0

≤

a

b

-

q

=

r

b

<

1

,
and for

b

<

0

0

≤

a

|

b

|

-

⌊

a

|

b

|

⌋

=

a

|

b

|

+

q

=

r

|

b

|

<

1

,

and the statement follows immediately.
∎

Proof of theorem 2.

Let

R

be a

commutative ring

with 1, and take

b

⁢

(

x

)

from

R

⁢

[

x

]

, where the

leading coefficient

of

b

⁢

(

x

)

is a unit in

R

. Without loss of generality we may assume the leading coefficient of

b

⁢

(

x

)

is 1.

If

n

is the degree of

b

⁢

(

x

)

, then set

q

⁢

(

x

)

=

{

0

if

deg

⁡

(

a

⁢

(

x

)

)

<

n

a

n

if

deg

⁡

(

a

⁢

(

x

)

)

=

n

,

where

a

n

is the leading coefficient of

a

⁢

(

x

)

. Then

r

⁢

(

x

)

=

a

⁢

(

x

)

-

q

⁢

(

x

)

⋅

b

⁢

(

x

)

is either 0 or

deg

⁡

(

r

⁢

(

x

)

)

<

deg

⁡

(

b

⁢

(

x

)

)

, as desired.

Now let

m

≥

deg

⁡

(

b

⁢

(

x

)

)

. Then the degree of the

polynomial

a

ˇ

⁢

(

x

)

=

a

⁢

(

x

)

-

a

m

+

1

⁢

b

⁢

(

x

)

⋅

x

m

+

1

-

n

is at most

m

. So by assumption we can write

a

⁢

(

x

)

as

a

⁢

(

x

)

=

b

⁢

(

x

)

⋅

(

q

ˇ

⁢

(

x

)

+

a

m

+

1

⁢

x

m

+

1

-

n

)

+

r

ˇ

⁢

(

x

)

where

r

ˇ

⁢

(

x

)

is either 0, or its degree is

<

b

⁢

(

x

)

.
∎

Title

proof of long division

Canonical

name

ProofOfLongDivision

Date of creation

2013-03-22 15:36:00

Last modified on

2013-03-22 15:36:00

Owner

Thomas Heye (1234)

Last modified by

Thomas Heye (1234)

Numerical id

6

Author

Thomas Heye (1234)

Entry type

Proof

Classification

msc 11A05

Classification

msc 12E99

Classification

msc 00A05

Generated on Fri Feb 9 15:10:30 2018 by

LaTeXML
Python code to calculate Euclid’s coefficients

Python code to calculate Euclid’s coefficients

Python code for calculating Euclid’s coefficients

\PMlinkescapetext{
def ExtGCD(x,y):
m,n=x,y
b,d=0,1
p,t=1,0
while (m!=0):
q = n//m # yes, //
n,m=m,n-q*m
d,b=b,d-q*b
t,p=p,t-q*p
return (t*x+d*y,t,d)

print ExtGCD(523,541) # Example from the parent entry
}

and it displays

\PMlinkescapetext{
>>> (1, 30, -29)
}

since

1

=

t

⁢

x

+

d

⁢

y

=

30

⋅

523

-

29

⋅

541

Title

Python code to

calculate

Euclid’s coefficients

Canonical name

PythonCodeToCalculateEuclidsCoefficients

Date of creation

2013-03-22 15:15:31

Last modified on

2013-03-22 15:15:31

Owner

drini (3)

Last modified by

drini (3)

Numerical id

7

Author

drini (3)

Entry type

Algorithm

Classification

msc 11A05

Generated on Fri Feb 9 15:10:31 2018 by

LaTeXML
quasiperfect number

quasiperfect number

If there exists an

abundant number

n

with

divisors

d

1

,

…

,

d

k

, such that

∑

i

=

1

k

d

i

=

2

⁢

n

+

1

,

that number would be called a

quasiperfect number

. Such a number would be

n

>

10

35

and have

ω

⁢

(

n

)

>

6

(where

ω

is the

number of distinct prime factors function

).

A quasiperfect number would thus overshoot the mark for being a

perfect number

by a margin of just 1. (The powers of 2 are short of perfection by a margin of 1).

Title

quasiperfect number

Canonical name

QuasiperfectNumber

Date of creation

2013-03-22 16:05:51

Last modified on

2013-03-22 16:05:51

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

6

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Related topic

AlmostPerfectNumber

Generated on Fri Feb 9 15:10:33 2018 by

LaTeXML
sandbox-maxima-worked-example-1

sandbox-maxima-worked-example-1

Here is some text which opens the discussion.
I’ll find out stuff later.
Third line in 1st part.
More text here. I wonder if it makes a separate paragraph.
2nd line in 2nd part.
First formula. Something about random blurfage.
2nd line in 3rd part.

eqn : x^2 + 17*x + 29 = 0;

x

2

+

17

⁢

x

+

29

=

0

solve (eqn, x);

[

x

=

-

173

+

17

2

,

x

=

173

-

17

2

]

subst (first (%), eqn);

(

173

+

17

)

2

4

-

17

⁢

(

173

+

17

)

2

+

29

=

0

expand (%);

0

=

0

More text here. I wonder if percent sign stuff works.

abc : '(def + xyz);

𝑥𝑦𝑧

+

𝑑𝑒𝑓

x

⁢

y

⁢

z

+

d

⁢

e

⁢

f

x

⁢

y

⁢

z

+

d

⁢

e

⁢

f

bcd : '(sin(x)+sin(x))$

Dollar sign terminator on previous line. Hmm.

ev (bcd);

2

⁢

sin

⁡

x

Now bcd is simplified.

print(bcd);

2

⁢

sin

⁡

x

bcd was simplified by printing too.
All finished.

No, really!

Title

sandbox-maxima-worked-example-1

Canonical

name

Sandboxmaximaworkedexample1

Date of creation

2014-12-31 17:36:26

Last modified on

2014-12-31 17:36:26

Owner

robert_dodier (1000903)

Last modified by

robert_dodier (1000903)

Numerical id

2

Author

robert_dodier (1000903)

Entry type

Example

Generated on Fri Feb 9 15:10:35 2018 by

LaTeXML
sandbox-maxima-worked-example-2

sandbox-maxima-worked-example-2

Converting inches to kilometers.
First load the ‘ezunits’ package.

load (ezunits) $

foo : 1 ` inch;

1

⁢

inch

Units are converted by the double backtick operator; conversions are not carried out automatically.

bar : foo `` km;

127

5000000

⁢

km

As usual, Maxima prefers exact numbers to inexact.
We can get a float

approximation

if we wish.

float (bar);

2.54

×

10

-

5

⁢

km

A volume conversion.
Here ‘baz’ is the volume of one inch of water spread over one acre, while ‘quux’ is the same volume expressed in SI units.

baz : 1 ` (inch * acre);

1

⁢

inch

⁢

acre

quux : baz `` m^3;

20076201783

195312500

⁢

m

3

float (%);

102.79015312896

⁢

m

3

Title

sandbox-maxima-worked-example-2

Canonical name

Sandboxmaximaworkedexample2

Date of creation

2013-12-27 13:29:32

Last modified on

2013-12-27 13:29:32

Owner

robert_dodier (1000903)

Last modified by

robert_dodier (1000903)

Numerical id

1

Author

robert_dodier (1000903)

Entry type

Example

Generated on Fri Feb 9 15:10:36 2018 by

LaTeXML
smooth number

smooth number

A

k

-

smooth number

n

is an

integer

such that its

prime divisors

are less than

k

. For example, 14824161510814728 is 7-smooth since its

prime factors

are just 2 and 3 (and thus its

divisors

are all either powers of those two primes or

multiples

of 6).

For small

k

, the number of smooth numbers less than a given

x

can be estimated with the formula

1

π

⁢

(

k

)

!

⁢

∏

i

=

1

π

⁢

(

k

)

log

⁡

x

log

⁡

p

i

,

where

π

⁢

(

x

)

is the

prime counting function

and

p

i

is the

i

th prime.

Smooth numbers have many applications, such as the many classic factorization algorithms which specifically call for smooth numbers in their initialization steps, as well as fast Fourier transform algorithms that also require smooth numbers.

Title

smooth number

Canonical name

SmoothNumber

Date of creation

2013-03-22 13:20:25

Last modified on

2013-03-22 13:20:25

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Related topic

RoughNumber

Generated on Fri Feb 9 15:10:40 2018 by

LaTeXML
sphenic number

sphenic number

Given three primes

p

<

q

<

r

, the

composite

integer

p

⁢

q

⁢

r

is a

sphenic number

. The first few sphenic numbers are 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154,

…

listed in A007304 of Sloane’s OEIS.

The

divisors

of a sphenic number therefore are

1

,

p

,

q

,

r

,

p

⁢

q

,

p

⁢

r

,

q

⁢

r

,

p

⁢

q

⁢

r

. Furthermore,

μ

⁢

(

p

⁢

q

⁢

r

)

=

(

-

1

)

3

(where

μ

is the Möbius

function

),

τ

⁢

(

p

⁢

q

⁢

r

)

=

8

(where

τ

is the

divisor function

) and

Ω

⁢

(

p

⁢

q

⁢

r

)

=

ω

⁢

(

p

⁢

q

⁢

r

)

=

3

(where

Ω

and

ω

are the number of (nondistinct)

prime factors

function (

http://planetmath.org/NumberOfNondistinctPrimeFactorsFunction

) and the

number of distinct prime factors function

, respectively).

The largest known sphenic number at any time is usually the product of the three largest known

Mersenne primes

.

Title

sphenic number

Canonical name

SphenicNumber

Date of creation

2013-03-22 16:10:33

Last modified on

2013-03-22 16:10:33

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

7

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A05

Generated on Fri Feb 9 15:10:42 2018 by

LaTeXML
Sturm’s theorem

Sturm’s theorem

This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.

Definition 1

.

Let

P

⁢

(

x

)

be a real polynomial in

x

, and define the

Sturm sequence

of

polynomials

(

P

0

⁢

(

x

)

,

P

1

⁢

(

x

)

,

…

)

by

P

0

⁢

(

x

)

=

P

⁢

(

x

)

P

1

⁢

(

x

)

=

P

′

⁢

(

x

)

P

n

⁢

(

x

)

=

-

rem

⁢

(

P

n

-

2

,

P

n

-

1

)

,

n

≥

2

Here

rem

⁢

(

P

n

-

2

,

P

n

-

1

)

denotes the

remainder

of the
polynomial

P

n

-

2

upon

division

by the polynomial

P

n

-

1

. The

sequence

terminates once one of the

P

i

is zero.

Definition 2

.

For any number

t

, let

v

⁢

a

⁢

r

P

⁢

(

t

)

denote the number of sign changes in the
sequence

P

0

⁢

(

t

)

,

P

1

⁢

(

t

)

,

…

.

Theorem 1

.

For

real numbers

a

and

b

that are both not roots of

P

⁢

(

x

)

,

#

⁢

{

distinct real roots of

⁢

P

⁢

in

⁢

(

a

,

b

)

}

=

v

⁢

a

⁢

r

P

⁢

(

a

)

-

v

⁢

a

⁢

r

P

⁢

(

b

)

In particular, we can count the
total number of distinct real roots by looking at the limits as

a

→

-

∞

and

b

→

+

∞

. The total number of
distinct real roots will depend only on the leading terms of the
Sturm sequence polynomials.

Note that deg

P

n

<

deg

P

n

-

1

, and so the longest possible
Sturm sequence has deg

P

+

1

terms.

Also, note that this sequence is very closely related to the sequence of remainders generated by the

Euclidean Algorithm

; in fact, the term

P

i

is the exact same except with a sign changed when

i

≡

2

or

3

(

mod

4

)

. Thus, the

Half-GCD Algorithm

may be used to compute this sequence. Be aware that some

computer algebra systems

may normalize remainders from the Euclidean Algorithm which messes up the sign.

For a proof, see Wolpert, N., “

http://web.archive.org/web/20050412175929/http://www.mpi-sb.mpg.de/ nicola/Vorlesung/sturm.ps

Proof of Sturm’s Theorem”

Title

Sturm’s theorem

Canonical name

SturmsTheorem

Date of creation

2013-03-22 14:28:49

Last modified on

2013-03-22 14:28:49

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

14

Author

rspuzio (6075)

Entry type

Theorem

Classification

msc 11A05

Classification

msc 26A06

Related topic

EuclidsAlgorithm

Defines

Sturm Sequence

Generated on Fri Feb 9 15:10:47 2018 by

LaTeXML
table of small multiply perfect numbers

table of small multiply perfect numbers

The first five

k

-

multiperfect numbers

for

1

<

k

<

6

are small enough to fit on a single page without the aid of horizontal scrollbars, or breaking up the numbers into more than one lines. For

k

>

5

, the numbers get quite big and writing them out becomes less practical. But since they tend to be

divisible

by lots of small primes, it makes sense to take advantage of the

primorials

. In the following table, the notation

n

⁢

#

means the

product

of the first

n

primes, while the notation

k

⁢

-

⁢

P

i

refers to the

i

th

k

-

perfect number

.

2

6

28

496

8128

33550336

3

120

672

523776

459818240

11

⁢

(

2

27

-

2

13

)

4

30240

32760

2178540

23569920

45532800

5

14182439040

31998395520

518666803200

13661860101120

30823866178560

6

297581328

⁢

(

5

⁢

-

⁢

P

3

)

(

1845

31

)

⁢

6

⁢

-

⁢

P

1

(

27335

369

)

⁢

6

⁢

-

⁢

P

2

(

13

⁢

#

)

⁢

210

18

⁢

412057

3

9

⁢

5

1

⁢

67

1

⁢

718241

(

13

⁢

#

)

⁢

2

18

⁢

105

5

⁢

793

58339155

The smallest 7-multiperfect number is 14131089794743834825984940273848552326434354481
8565120000.

The source for

1

<

k

<

6

in the table are the following

sequences

in Sloane’s OEIS: A000396, A005820, A027687, A046060 and A046061. These have all been verified with Mathematica. For larger

k

, the information comes from the

Multiply Perfect Numbers Page

but has not been doublechecked anew, as these numbers require far more intensive computational effort to verify.

Title

table of small multiply perfect numbers

Canonical name

TableOfSmallMultiplyPerfectNumbers

Date of creation

2013-03-22 17:48:21

Last modified on

2013-03-22 17:48:21

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

16

Author

PrimeFan (13766)

Entry type

Data Structure

Classification

msc 11A05

Generated on Fri Feb 9 15:10:49 2018 by

LaTeXML
Thabit number

Thabit number

An

integer

of the form

3

⋅

2

n

-

1

, or

2

n

+

1

+

2

n

-

1

. They are listed in A055010 of Sloane’s OEIS. The

Thabit numbers

are a subset of the

Proth numbers

.

The mathematician and astronomer

Thabit ibn Qurra

studied these numbers in search of a

formula for amicable pairs

. He found that when two consecutive Thabit numbers are also

prime numbers

(corresponding to indices

n

and

n

-

1

) and

9

⋅

2

2

⁢

n

-

1

-

1

is a prime number, too, then these numbers multiplied by

2

n

will reveal an

amicable pair

. The only

n

known to fit these criteria are 2, 4 and 7. The largest Thabit number known to be prime corresponds to index 2312734, its immediate lower neighbor is

composite

.

It is conjectured that the nimfactorial of a Thabit number is always 2.

Title

Thabit number

Canonical name

ThabitNumber

Date of creation

2013-03-22 15:52:58

Last modified on

2013-03-22 15:52:58

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

5

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 11A05

Synonym

Thabit ibn Kurra number

Synonym

Thabit ibn Kurrah number

Synonym

Thabit ibn Qurra number

Synonym

Thabit ibn Qurrah number

Synonym

Thabit bin Kurra number

Synonym

Thabit bin Kurrah number

Synonym

Thabit bin Qurra number

Synonym

Thabit bin Qurrah number

Related topic

AFormulaForAmicablePairs

Generated on Fri Feb 9 15:10:50 2018 by

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