aarjaneiro/planetmath huggingface dataset:
Smarandache-Wellin prime

Smarandache-Wellin prime

An

integer

that is both a

Smarandache-Wellin number

in a given base

b

and a

prime number

.

References

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001: 72

H. Ibstedt, A Few

Smarandache Sequences

,

Smarandache Notions Journal

, Vol. 8, No. 1-2-3, 1997: 170 - 183

Title

Smarandache-Wellin prime

Canonical name

SmarandacheWellinPrime

Date of creation

2013-03-22 15:54:40

Last modified on

2013-03-22 15:54:40

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

6

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Related topic

FlorentinSmarandache

Generated on Fri Feb 9 15:23:58 2018 by

LaTeXML
Smith number

Smith number

A

Smith number

is a number which in a given base, the sum of its digits is equal to the sum of the digits in its factorization. (In the case of numbers that are not

square-free

, the factorization is written without exponents, writing the repeated factor as many times as needed). For example, 202 is a Smith number, since 2 + 0 + 2 = 4, and its factorization is

2

×

101

, and 2 + 1 + 0 + 1 = 4.

Prime numbers

are not considered, since it is

obvious

that all of them satisfy the condition given above.

In base 10, the first few Smith numbers are

4, 22, 27, 58, 85, 94, 121, 166, 202, 265, 274, 319, 346, 355, 378, 382, 391, 438, 454, 483 (

sequence

A006753 in the OEIS).

There are infinitely many Smith numbers that are also

palindromic numbers

.

Smith numbers were named by Albert Wilansky of Lehigh University for his brother-in-law Harold Smith whose phone number (493-7775) was the first noticed Smith number.

Smith numbers can be constructed from factored

repunits

. The largest known Smith number is (as of 2005)

9

×

R

1031

×

(

10

4594

+

3

×

10

2297

+

1

)

1476

×

10

3913210

where

R

1031

=

2

1032

-

1

.

Title

Smith number

Canonical

name

SmithNumber

Date of creation

2013-03-22 16:19:32

Last modified on

2013-03-22 16:19:32

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:23:59 2018 by

LaTeXML
some divisibility tests

some divisibility tests

We want to derive the

divisibility tests

for 3, 9, and 11.

Let

A

=

a

n

⋅

10

n

+

a

n

-

1

⋅

10

n

-

1

+

…

+

a

0

be the digital

representation

of the

integer

A

in the

decadic

system (

0

≤

a

i

≤

9

).

Since

10

≡

1

(

mod

3

,

9

)

,  we have also

10

i

≡

1

(

mod

3

,

9

)

for each

i

,  and hence

a

i

⋅

10

i

≡

a

i

(

mod

3

,

9

)

.

Consequently

A

=

∑

i

=

0

n

a

i

⋅

10

i

≡

∑

i

=

0

n

a

i

(

mod

3

,

9

)

.

This

congruence

means that

∑

i

=

0

n

a

i

≡

0

(

mod

3

,

9

)

⇔

A

≡

0

(

mod

3

,

9

)

.

(1)

The fact

10

≡

-

1

(

mod

11

)

similarly yields

A

=

∑

i

=

0

n

a

i

⋅

10

i

≡

∑

i

=

0

n

(

-

1

)

i

⁢

a

i

(

mod

11

)

,

whence

∑

i

=

0

n

(

-

1

)

i

⁢

a

i

≡

0

(

mod

11

)

⇔

A

≡

0

(

mod

11

)

.

(2)

The results (1) and (2) may be rendered as the following:

•

A

is

divisible

by 3 (resp. 9) if and only if

a

0

+

a

1

+

…

+

a

n

is.

•

A

is divisible by 11 if and only if

a

0

-

a

1

+

-

…

+

(

-

1

)

n

a

n

is.

Title

some divisibility tests

Canonical name

SomeDivisibilityTests

Date of creation

2014-05-29 11:26:48

Last modified on

2014-05-29 11:26:48

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

7

Author

pahio (2872)

Entry type

Derivation

Classification

msc 11A63

Classification

msc 11A07

Generated on Fri Feb 9 15:24:02 2018 by

LaTeXML
strictly non-palindromic number

strictly non-palindromic number

If for a given

integer

n

>

0

there is no base

1

<

b

<

(

n

-

1

)

such that each digit

d

i

=

d

k

+

1

-

i

of

n

(where

k

is the number of

significant digits

of

n

in base

b

and

i

is a simple iterator in the range

0

<

i

<

(

k

+

1

)

), meaning that

n

is not a

palindromic number

in any of these bases, then

n

is called a

strictly non-palindromic number

.

Clearly

n

>

2

will be palindromic for

b

=

n

-

1

, and though trivially, this is also true for

b

>

n

.

6 is the largest

composite

strictly non-palindromic number. For any other

2

|

n

, it is easy to find a base in which

n

is written

22

b

by simply computing

b

=

n

2

-

1

. For odd composites

n

=

m

⁢

p

, where

p

is an

odd prime

and

m

≥

p

we can almost always either find that for

b

=

p

-

1

,

n

=

b

2

+

2

⁢

b

+

1

, or for

b

=

m

-

1

then

n

=

p

⁢

b

+

p

and written with two instances of the digit corresponding to

p

in that base. The one odd case of

n

=

9

is quickly dismissed with

b

=

2

.

Title

strictly non-palindromic number

Canonical name

StrictlyNonpalindromicNumber

Date of creation

2013-03-22 16:25:12

Last modified on

2013-03-22 16:25:12

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:24:06 2018 by

LaTeXML
strobogrammatic number

strobogrammatic number

A

strobogrammatic number

is a number that, given a base and given a set of glyphs, appears the same whether viewed normally or upside down. In base 10, given a set of glyphs where 0, 1 and 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other upside down (such as the digit

characters

in ASCII using the font Stylus BT), the first few strobogrammatic numbers are:

1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001 (

sequence

A000787 in OEIS)

Like the concept of

repunits

and

palindromic numbers

, the concept of strobogrammatic numbers is base-dependent. Unlike palindromicity it is also font dependent. But the concept of strobogrammatic numbers is not neatly expressible algebraically, the way that the concept of repunits is, or even the concept of palindromic numbers.

There are sets of glyphs for writing numbers in base 10, such as the Devanagari and Gurmukhi of India in which the numbers listed above are not strobogrammatic at all.

In binary, given a glyph for 1 consisting of a single line without hooks or serifs, all palindromic numbers are strobogrammatic, which means that all

Mersenne numbers

are strobogrammatic.

Title

strobogrammatic number

Canonical name

StrobogrammaticNumber

Date of creation

2013-03-22 16:46:35

Last modified on

2013-03-22 16:46:35

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:24:08 2018 by

LaTeXML
strobogrammatic prime

strobogrammatic prime

A

strobogrammatic prime

is a

prime number

that, given a base and given a set of glyphs, is a

strobogrammatic number

(it appears the same whether viewed normally or upside down). In base 10, given a set of glyphs where 0, 1 and 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other upside down, (such as on the

seven-segment display

of a

calculator

), the first few strobogrammatic primes are: 11, 101, 181, 619, 16091, 18181 (listed in A007597 of Sloane’s OEIS).

In binary, given a glyph for 1 consisting of a single line without hooks or serifs, all

Mersenne primes

are strobogrammatic primes.

Palindromic primes

in binary are also strobogrammatic.

In base 10,

dihedral primes

that don’t use 2 or 5 are also strobogrammatic primes.

Title

strobogrammatic prime

Canonical

name

StrobogrammaticPrime

Date of creation

2013-03-22 16:19:24

Last modified on

2013-03-22 16:19:24

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

5

Author

PrimeFan (13766)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:24:09 2018 by

LaTeXML
sturdy number

sturdy number

A

sturdy number

is a

positive integer

n

having a given binary weight (number of on bits, or

instances

of the digit 1 in its binary

representation

) such that no

multiple

m

⁢

n

(with

m

being any positive integer) has a lesser binary weight. For example, 10 is written 1010 in binary; it has two on bits; any multiple of 10 has at least two on bits.

All

powers of two

corresponding to positive integer

exponents

are sturdy numbers, since its representation can only have one on bit (the one corresponding to the appropriate power of two), and multiplying by another power of two gives yet another power of two, while multiplying by any other

integer

gives a number with at least two on bits.

Likewise we can verify that integers having two on bits are also sturdy numbers, since none of its multiples can be a power of two and therefore any multiple must also have at least two on bits.

A125121 in Sloane’s OEIS gives the following list of the first sixty-eight sturdy numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 20, 21, 24, 28, 30, 31, 32, 33, 34, 35, 36, 40, 42, 45, 48, 49, 51, 56, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 73, 75, 80, 84, 85, 89, 90, 93, 96, 98, 102, 105, 112, 120, 124, 126, 127, 128, 129, 130, 132, 133, 135, 136. However, these were computed with the

assumption

that it was only necessary to test up to

m

=

14

×

10

7

.

Title

sturdy number

Canonical name

SturdyNumber

Date of creation

2013-03-22 19:08:07

Last modified on

2013-03-22 19:08:07

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:24:11 2018 by

LaTeXML
sum-product number

sum-product number

A

sum-product number

is an

integer

n

that in a given base satisfies the equality

n

=

∑

i

=

1

m

d

i

⁢

∏

i

=

1

m

d

i

where

d

i

is a digit of

n

, and

m

is the number of digits of

n

. This means a test of whether the sum of the digits of

n

times the product of the digits of

n

is equal to

n

.

For example, the number 128 in base 7 is a sum-product number since

242

7

=

(

2

+

4

+

2

)

⁢

(

2

⋅

4

⋅

2

)

All sum-product numbers are

Harshad numbers

, too.

0 and 1 are sum-product numbers in any positional base. The proof that the set of sum-product numbers in base 2 is finite is elementary enough not to inspire claims of authorship. The proof that the set of sum-product numbers in base 10 is finite (specifically, 0, 1, 135 and 144) is more involved but within the realm of

basic algebra

, and it points the way to a formulation of the

proof that number of sum-product numbers in any base is finite

.

Title

sum-product number

Canonical name

SumproductNumber

Date of creation

2013-03-22 15:46:50

Last modified on

2013-03-22 15:46:50

Owner

Mravinci (12996)

Last modified by

Mravinci (12996)

Numerical id

8

Author

Mravinci (12996)

Entry type

Definition

Classification

msc 11A63

Synonym

sum product number

Generated on Fri Feb 9 15:24:12 2018 by

LaTeXML
surd

surd

In mathematics, numbers often have to be written using roots or fractional exponents. For example, the number

2

cannot be written without using a root or writing

2

1

/

2

. A

surd

is an

irrational number

which can be written as the sum of

rational

powers of rational

numbers.
Another example would be

3

⁢

2

-

5

3

.

References

http://www.mathcentre.ac.uk/students.php/all_subjects/

arithmetic

/surds/resources/34

What is a surd?

Title

surd

Canonical

name

Surd

Date of creation

2013-03-22 14:17:44

Last modified on

2013-03-22 14:17:44

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

10

Author

mathwizard (128)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:24:14 2018 by

LaTeXML
trimorphic number

trimorphic number

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’s also the case that the

k

least significant digits of

n

3

are the same of those of

n

, then

n

is called a

trimorphic number

.

All

automorphic numbers

(with

m

=

1

) are also trimorphic numbers, but not all trimorphic numbers are 1-automorphic.

Title

trimorphic number

Canonical name

TrimorphicNumber

Date of creation

2013-03-22 16:21:32

Last modified on

2013-03-22 16:21:32

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

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

LaTeXML
vampire number

vampire number

Consider the

integer

1395. In the equation

1395

=

15

⋅

93

,

expressed in base 10, both sides (

http://

planetmath

.org/Equation

) use the same digits.

When a number with an

even number

of digits is also the product of two multiplicands having half as many digits as the product, and together having the same digits, the product is called a

vampire number

. The multiplicands are called

fangs

.

By definition, a vampire number can’t be a

prime number

. But if both of its fangs are prime numbers, then it might be referred to as a “prime vampire number.”

This concept can be applied to any positional base, and to

Roman numerals

. For example,

V

⁢

I

⁢

I

⁢

I

=

I

⁢

I

⋅

I

⁢

V

.

A vampire number is automatically a

Friedman number

also.

Title

vampire number

Canonical name

VampireNumber

Date of creation

2013-03-22 15:45:10

Last modified on

2013-03-22 15:45:10

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

8

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Defines

vampire number

Defines

fang

Generated on Fri Feb 9 15:24:25 2018 by

LaTeXML
Zeckendorf’s theorem

Zeckendorf’s theorem

Theorem

. Every

positive integer

can be represented as a sum of distinct non-consecutive

Fibonacci numbers

in a unique way.

This is

Zeckendorf’s theorem

, first formulated by Edouard Zeckendorf.

For our purposes here, define the

Fibonacci sequence

thus:

F

0

=

1

,

F

1

=

1

and

F

m

=

F

m

-

2

+

F

m

-

1

for all

m

>

0

. 1 and 1 are not distinct even though the first is

F

0

and the latter is

F

1

. We will consider two Fibonacci numbers

F

i

and

F

j

consecutive

if their indexes

i

and

j

are consecutive

integers

, e.g.,

j

=

i

+

1

.

A

consequence

of the theorem is that for every positive integer

n

there is a unique

ordered tuplet

Z

consisting of

k

elements

, all 0s or 1s, such that

∑

i

=

1

k

Z

i

⁢

F

i

=

n

,

where

Z

i

is the

i

th element in

Z

. This ordered tuplet

Z

is the

Zeckendorf representation

of

n

, or we might even say the

Fibonacci base

representation of

n

(or the

Fibonacci coding

of

n

).

So for example, 53 = 34 + 13 + 5 + 1, that is,

F

8

+

F

6

+

F

4

+

F

1

. Furthermore,

Z

=

(

1

,

0

,

1

,

0

,

1

,

0

,

0

,

1

)

. We list the constituent elements in

descending order

from

Z

k

to

Z

1

to facilitate reinterpretation as a binary integer, 10101001 (or 169) in this example. Taking the Zeckendorf representations of integers in order and reinterpreting in binary as

∑

i

=

1

k

Z

i

⁢

2

i

-

1

gives the

sequence

1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, 21, 32, 33, 34, … (A003714 in Sloane’s OEIS). It can be observed that these numbers have no consecutive 1s in their binary representations.

References

1

J. Tatersall,

Elementary number theory

in nine chapters

Cambridge: Cambridge University Press (2005): 44

2

J.-P. Allouche, J. Shallit and G. Skordev, “Self-generating sets, integers with missing blocks and substitutions”

Discrete Math

.

, 292 (2005): 1 - 15

Title

Zeckendorf’s theorem

Canonical name

ZeckendorfsTheorem

Date of creation

2013-03-22 16:03:57

Last modified on

2013-03-22 16:03:57

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

11

Author

CompositeFan (12809)

Entry type

Theorem

Classification

msc 11A63

Classification

msc 11B39

Synonym

Zeckendorff’s theorem

Related topic

FibonacciSequence

Related topic

UniquenessOfDigitalRepresentation

Defines

Zeckendorf representation

Defines

Fibonacci base

Defines

Fibonacci coding

Generated on Fri Feb 9 15:24:27 2018 by

LaTeXML
Zuckerman number

Zuckerman number

Consider the

integer

384. Multiplying its digits,

3

×

8

×

4

=

96

and

384

96

=

91

.

When an integer is

divisible

by the product of its digits, it’s called a

Zuckerman number

. That is, given

m

is the number of digits of

n

and

d

x

(for

x

≤

k

) is an integer of

n

,

∏

i

=

1

m

d

i

|

n

All 1-digit numbers and the base number itself are Zuckerman numbers.

It is possible for an integer to be divisible by its multiplicative digital root and yet not be a Zuckerman number because it doesn’t divide its first digit product evenly (for example, 1728 in base 10 has multiplicative digital root 2 but is not divisible by

1

×

7

×

2

×

8

=

112

). The reverse is also possible (for example, 384 is divisible by 96, as shown above, but clearly not by its multiplicative digital root 0).

References

1

J. J. Tattersall,

Elementary number theory

in nine chapters

, p. 86. Cambridge: Cambridge University Press (2005)

Title

Zuckerman number

Canonical name

ZuckermanNumber

Date of creation

2013-03-22 16:04:36

Last modified on

2013-03-22 16:04:36

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

4

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 11A63

Generated on Fri Feb 9 15:24:29 2018 by

LaTeXML
adjacent fraction

adjacent fraction

Two

fractions

a

b

and

c

d

,

a

b

>

c

d

of the

positive integers

a

,

b

,

c

,

d

are

adjacent

if their

difference

is some

unit fraction

1

n

,

n

>

0

that is, if we can write:

a

b

-

c

d

=

1

n

.

For example the two proper fractions and unit fractions

1

11

and

1

12

are adjacent since:

1

11

-

1

12

=

1

132

.

1

17

and

1

19

are not since:

1

17

-

1

19

=

2

323

.

It is not necessary of course that fractions are both proper fractions:

20

19

-

19

19

=

1

19

.

or unit fractions:

3

4

-

2

3

=

1

12

.

All successive terms of some

Farey sequence

F

n

of a degree

n

are always

adjacent fractions

. In the first Farey sequence

F

1

of a degree 1 there are only two adjacent fractions, namely

1

1

and

0

1

.

Adjacent unit fractions can be parts of many

Egyptian fractions

:

1

70

+

1

71

=

141

4970

.

Title

adjacent fraction

Canonical

name

AdjacentFraction

Date of creation

2013-03-22 12:48:23

Last modified on

2013-03-22 12:48:23

Owner

XJamRastafire (349)

Last modified by

XJamRastafire (349)

Numerical id

17

Author

XJamRastafire (349)

Entry type

Definition

Classification

msc 11A67

Related topic

FareySequence

Related topic

UnitFraction

Related topic

ContinuedFraction

Related topic

NumeratorAndDenominatorIncreasedBySameAmount

Generated on Fri Feb 9 15:24:31 2018 by

LaTeXML
any rational number is a sum of unit fractions

any rational number is a sum of unit fractions

Representation

Any

rational number

a

b

∈

ℚ

between 0 and 1 can be represented as a sum of different

unit fractions

. This result was known to the Egyptians, whose way for representing rational numbers was as a sum of different unit fractions.

The following greedy algorithm can represent any

0

≤

a

b

<

1

as such a sum:

1.

Let

n

=

⌈

b

a

⌉

be the smallest

natural number

for which

1

n

≤

a

b

. If

a

=

0

, terminate.

2.

Output

1

n

as the next term of the sum.

3.

Continue from step 1, setting

a

′

b

′

=

a

b

-

1

n

.

Proof of correctness

Proof.

The algorithm can never output the same unit fraction twice. Indeed, any

n

selected in step 1 is at least 2, so

1

n

-

1

<

2

n

– so the same

n

cannot be selected twice by the algorithm, as then

n

-

1

could have been selected instead of

n

.

It remains to prove that the algorithm terminates. We do this by

induction

on

a

.

For

a

=

0

:

The algorithm terminates immediately.

For

a

>

0

:

The

n

selected in step 1

satisfies

b

≤

a

⁢

n

<

b

+

a

.

So

a

b

-

1

n

=

a

⁢

n

-

b

b

⁢

n

,

and

0

≤

a

⁢

n

-

b

<

a

– by the

induction hypothesis

, the algorithm terminates for

a

b

-

1

n

.

∎

Problems

1.

The greedy algorithm always works, but it tends to produce unnecessarily large

denominators

. For

instance

,

47

60

=

1

3

+

1

4

+

1

5

,

but the greedy algorithm selects

1

2

, leading to the representation

47

60

=

1

2

+

1

4

+

1

30

.

2.

The representation is

never

unique. For instance, for any

n

we have the representations

1

n

=

1

n

+

1

+

1

n

⋅

(

n

+

1

)

So given any one representation of

a

b

as a sum of different unit fractions we can take the largest denominator appearing

n

and replace it with two (larger) denominators. Continuing the process indefinitely, we see infinitely many such representations, always.

Title

any rational number is a sum of unit fractions

Canonical

name

AnyRationalNumberIsASumOfUnitFractions

Date of creation

2013-03-22 12:48:31

Last modified on

2013-03-22 12:48:31

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

7

Author

Mathprof (13753)

Entry type

Derivation

Classification

msc 11A67

Synonym

Egyptian algorithm

Generated on Fri Feb 9 15:24:33 2018 by

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Erdős-Straus conjecture

Erdős-Straus conjecture

In 1948, Paul Erdős and Ernst Straus conjectured that for an

integer

n

>

1

there is always a solution to

4

n

=

1

a

+

1

b

+

1

c

where

a

,

b

and

c

are integers in the relation

0

<

a

≤

b

≤

c

. This is the

Erdős-Straus

. Put another way,

4

n

can be rewritten as a sum of three

unit fractions

. The three unit fractions need not be distinct, but some people consider solutions with distinct unit fractions to be more elegant. It is believed that for all

n

>

4

solutions with distinct unit fractions are possible.

As with any

conjecture

, a single

counterexample

is enough to disprove, but no multitude of examples is enough to prove. With brute force

computer

calculations, Allan Swett has obtained examples for all

n

<

10

14

. Because of the

Hasse principle

of

Diophantine equations

, we can be sure that for

semiprimes

p

⁢

q

(where

p

and

q

are distinct primes) a solution can be found by looking at

4

p

or

4

q

. Researchers are therefore certain that if a counterexample exists, it is surely a

prime number

. Thus Swett has only made available the raw data only for selected prime

n

rather than for all

n

he tested.

Title

Erdős-Straus conjecture

Canonical

name

ErdHosStrausConjecture

Date of creation

2013-03-22 16:28:04

Last modified on

2013-03-22 16:28:04

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

8

Author

CompositeFan (12809)

Entry type

Conjecture

Classification

msc 11A67

Synonym

Erdös-Straus conjecture

Synonym

Erdos-Straus conjecture

Synonym

Erdos-Strauss conjecture

Generated on Fri Feb 9 15:24:35 2018 by

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