Matula Explorer

Matula Number Explorer

by William Sharkey

Welcome. To change this document, click the edit tab. You can enter a number range you are interested in, like one to six: ¹ ^{2
(
²
)} ^{3
(
^{4
(
³
)}
)} ^{4
(
³
)(
³
)} ^{5
(
^{6
(
^{6
(
⁵
)}
)}
)} ^{6
(
^{6
(
⁵
)}
)(
⁵
)} After editing, click the "Expore" tab to see the results. You can drag and drop circles to change the number-trees. Double click a circle to re-root the tree with the selected number. ~~~ David W. Matula discovered that you can associate an integer with each rooted tree. Google "Matula Number" for more. This site calculates the Matula number at each node of a given tree. Notice how the leaves of Matula trees are prime numbers, and the forks are composite numbers. For a given number of vertices (v) there are only so many unique trees that can be formed. I enumerated trees up to seven vertices. To inform this site to draw a Matula tree, I just enter a number into the edit pane. This site generates all of the other vertices for you. To re-root a tree, click on a number. To add a vertex, drag a circle from another tree. To get rid of a vertex, drag it out of the tree onto the page. To enumerate a range of trees, use the format a:b where a and b are the starting and stoping number, in the edit tab. ~~~ Matula trees by number of vertices: zero vertices -> one empty space ⁰ one vertice -> one tree ¹ two vertices -> one tree ^{2
(
²
)} three vertices -> one tree ^{3
(
^{4
(
³
)}
)} four vertices -> two trees ^{5
(
^{6
(
^{6
(
⁵
)}
)}
)} ^{7
(
^{8
(
⁷
)(
⁷
)}
)} five vertices -> three trees ^{11
(
^{10
(
^{9
(
^{10
(
¹¹
)}
)}
)}
)} ^{17
(
^{14
(
^{12
(
¹³
)(
¹³
)}
)}
)} ^{53
(
^{32
(
⁵³
)(
⁵³
)(
⁵³
)(
⁵³
)}
)} six vertices -> six trees ^{31
(
^{22
(
^{15
(
^{15
(
^{22
(
³¹
)}
)}
)}
)}
)} ^{41
(
^{26
(
^{18
(
^{26
(
⁴¹
)}
)(
²³
)}
)}
)} ^{43
(
^{28
(
^{28
(
⁴³
)(
⁴³
)}
)(
⁴³
)}
)} ^{59
(
^{34
(
^{21
(
^{20
(
²⁹
)(
²⁹
)}
)}
)}
)} ^{67
(
^{38
(
^{24
(
³⁷
)(
³⁷
)(
³⁷
)}
)}
)} ^{131
(
^{64
(
¹³¹
)(
¹³¹
)(
¹³¹
)(
¹³¹
)(
¹³¹
)}
)} seven vertices -> eleven trees ^{83
(
^{46
(
^{27
(
^{46
(
⁸³
)}
)(
^{46
(
⁸³
)}
)}
)}
)} ^{127
(
^{62
(
^{33
(
^{25
(
^{33
(
^{62
(
¹²⁷
)}
)}
)}
)}
)}
)} ^{131
(
^{64
(
¹³¹
)(
¹³¹
)(
¹³¹
)(
¹³¹
)(
¹³¹
)}
)} ^{139
(
^{68
(
^{49
(
^{68
(
¹³⁹
)(
¹³⁹
)}
)}
)(
¹³⁹
)}
)} ^{157
(
^{74
(
^{36
(
^{74
(
¹⁵⁷
)}
)(
⁶¹
)(
⁶¹
)}
)}
)} ^{163
(
^{76
(
^{56
(
¹⁰⁷
)(
¹⁰⁷
)(
¹⁰⁷
)}
)(
¹⁶³
)}
)} ^{179
(
^{82
(
^{39
(
^{30
(
^{58
(
¹⁰⁹
)}
)(
⁴⁷
)}
)}
)}
)} ^{191
(
^{86
(
^{42
(
^{52
(
¹⁰¹
)(
¹⁰¹
)}
)(
⁷³
)}
)}
)} ^{241
(
^{106
(
^{48
(
⁸⁹
)(
⁸⁹
)(
⁸⁹
)(
⁸⁹
)}
)}
)} ^{277
(
^{118
(
^{51
(
^{35
(
^{44
(
⁷⁹
)(
⁷⁹
)}
)}
)}
)}
)} ^{331
(
^{134
(
^{57
(
^{40
(
⁷¹
)(
⁷¹
)(
⁷¹
)}
)}
)}
)} It is easy to refer to trees and nodes with just a Matula number. No arbitrary labeling schemes are required (if you do not care to differentiate vertices that are equivalent under isomorphism). Conjoining two trees together at vertex p and q is the same as multiplying p and q's Matula number. Notice how composite numbers belonging to trees of vertex count v are limited to using prime numbers established at lower v trees? Can you prove this? What other things can we say about the average number of primes as a function of v? Trees can be entered as parenthesis, for example, two parenthesis sets next to each other is equal to ^{4
(
³
)(
³
)} . If you want to know what the nth prime number is, just surround n with parenthesis like this: (n). Here is the hundreth prime: ^{541
(
^{200
(
^{519
(
^{2062
(
⁸²²¹
)}
)}
)(
^{519
(
^{2062
(
⁸²²¹
)}
)}
)(
⁵⁴¹
)(
⁵⁴¹
)}
)} . Have Fun -- William

Welcome.

To change this document, click the edit tab.

You can enter a number range you are interested in, like one to six:

1:6

After editing, click the "Expore" tab to see the results.

You can drag and drop circles to change the number-trees.

Double click a circle to re-root the tree with the selected number.

~~~

David W. Matula discovered that you can associate an integer with each rooted tree. Google "Matula Number" for more.

This site calculates the Matula number at each node of a given tree.

Notice how the leaves of Matula trees are prime numbers, and the forks are composite numbers.

For a given number of vertices (v) there are only so many unique trees that can be formed.

I enumerated trees up to seven vertices.

To inform this site to draw a Matula tree, I just enter a number into the edit pane.

This site generates all of the other vertices for you.

To re-root a tree, click on a number. To add a vertex, drag a circle from another tree. To get rid of a vertex, drag it out of the tree onto the page.

To enumerate a range of trees, use the format a:b where a and b are the starting and stoping number, in the edit tab.

~~~

Matula trees by number of vertices:

zero vertices  -> one empty space
0

one vertice    -> one tree
1

two vertices   -> one tree
2

three vertices -> one tree
3

four vertices  -> two trees
5 7

five vertices  -> three trees
11 17 53

six vertices   -> six trees
31 41 43 
59 67 131

seven vertices -> eleven trees
83 127 131
139 157 163
179 191 241
277 331

It is easy to refer to trees and nodes with just a Matula number. No arbitrary labeling schemes are required (if you do not care to differentiate vertices that are equivalent under isomorphism).

Conjoining two trees together at vertex p and q is the same as multiplying p and q's Matula number.

Notice how composite numbers belonging to trees of vertex count v are limited to using prime numbers established at lower v trees? Can you prove this?

What other things can we say about the average number of primes as a function of v?

Trees can be entered as parenthesis, for example, two parenthesis sets next to each other is equal to ()().

If you want to know what the nth prime number is, just surround n with parenthesis like this: (n). Here is the hundreth prime: (100).

Have Fun
-- William

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