# Find the next number of the sequence 1, 3, 5, 7, _

## Full text

(1)

(2)
(3)
(4)

### HW1 b04701232 陳柔安

• Interesting sequence: look-and-say sequence

➢If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For d different from 1, the sequence starts as follows:

d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …

➢e.g. 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, ...

• Why interesting?

➢No digits other than 1, 2, and 3 appear in the sequence, unless the seed number contains such a digit or a run of more than three of the same digit.

➢The sequence grows indefinitely, except for the degenerate sequence: 22, 22, 22, 22, …

(5)
(6)

### f(1) = 1, f(2) = 3, f(3) = 5, f(4) = 7, f(5) = 217341

Reference: http://joyreactor.com/post/1666199

(7)

### 0.01234567890123456789012345 67890123456789...

• 這是我在網路上偶然看到的，我自己當下看到覺得還蠻有趣的

(8)
(9)
(10)

### 黑洞數 Kaprekar's constant

3321 – 1233 = 2088

8820 – 0288 = 8532

8532 – 2358 = 6174

7641 – 1467 = 6174

3321 – 1233 = 2088

8820 – 0288 = 8532

8532 – 2358 = 6174

7641 – 1467 = 6174

### 6174

6321 – 1236 = 5085

8550 – 0558 = 7992

9972 – 2799 = 7173

7731 – 1377 = 6354

6543 – 3456 = 3087

8730 – 0378 = 8352

8532 – 2358 = 6174

6321 – 1236 = 5085

8550 – 0558 = 7992

9972 – 2799 = 7173

7731 – 1377 = 6354

6543 – 3456 = 3087

8730 – 0378 = 8352

8532 – 2358 = 6174

981 – 189 = 792

972 – 279 = 693

963 – 369 = 594

954 – 459 = 495

981 – 189 = 792

972 – 279 = 693

963 – 369 = 594

954 – 459 = 495

620 – 026 = 594

954 – 459 = 495

620 – 026 = 594

954 – 459 = 495

(11)

### Truncatable Primes

• Left-truncatable primes (OEIS A024785):

• 2, 3, 5, 7, 13, 17, 23, 37, 43, 47, 53, 67, 73, 83, 97, 113, 137, 167,

173, 197, 223, 283, 313, 317, 337, 347, 353, 367, 373, 383, 397, 443, 467, 523, 547, 613, 617, 643, 647, 653, 673, 683, 743, 773, 797, 823, 853, 883, 937, 947, 953, 967, 983, 997, ... ,

357686312646216567629137

• Right-truncatable primes (OEIS A024770):

• 2, 3, 5, 7, 23, 29, 31, 37, 53, 59, 71, 73, 79, 233, 239, 293, 311, 313, 317, 373, 379, 593, 599, 719, 733, 739, 797, 2333, 2339, 2393, 2399, 2939, 3119, 3137, 3733, 3739, 3793, 3797, 5939, 7193, 7331, 7333, 7393, … , 73939133

• It’s interesting that these prime numbers (contain no zero) remain prime when the last “right/left” digit is successively

removed. For example, 7393 is a right-truncatable prime, since 7393, 739, 73, and 7 are all prime.

R08922041 資工碩一 張立暐

• Related: (truncatable)

### PRIME DAYs!

• 20190823 is prime

• 190823 is prime

• 90823 is prime

• 823 is prime

• 23 is prime

• 3 is prime

• There are only 53

(truncatable) prime days from 2000 to 2999.

20191001 is not prime. It is divisible by 139.

(12)

### Palindromic Prime

A palindromic prime is a prime number that is also a palindromic number.

The first few decimal palindrome primes are:

2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929 … and the biggest one found so far is

10474500 + 999 × 10237249 + 1

Except for 11, all palindromic primes have an odd number of digits, because all palindromes with even number of digits can be divided by 11.

Due to the superstitious significance of the numbers it contains, the palindromic prime 1000000000000066600000000000001 is known

as Belphegor's Prime with “666” in the center,  on either side enclosed by thirteen zeroes and a one. Coincidence? I don’t think so.

There are lots of examples of primes and palindromes, so I’m curious about what happens to a prime which is also a palindrome. I found it interesting, especially the Belphegor's Prime.

R08922049 資工碩一 林政豪

(13)

### Lazy caterer's sequence

Definition : The maximum pieces with given cuts

� (� )=

�+12

+1=

2

+

1

+

0

### )

= 2+2� +1

Image source from FAMOUS MATHEMATICAL SEQUENCES AND SERIES Proof can be referenced to Lazy caterer's sequence - Wikipedia

[hw1] R08922060 林映廷

(14)

### Khinchin's constant

Reference: Wikipedia, Numberphile In number theory, Aleksandr

Yakovlevich Khinchin proved that for almost all real numbers x,

coefficients ai of the continued fraction expansion of x have a finite geometric mean that is

independent of the value of x and is known as Khinchin's constant.

It is interesting that almost every numbers share a same value in terms of the geometric mean of the coefficients of the continued fraction expansion, and that K0 itself is thought to follow this rule as well.

R08922112 林溥博

(15)

### The n-th formula of them is

Dyck Paths. Courtesy to Dmharvey on Wikipedia

d05945019 楊宗儒

(16)

### Xie Bing Ang 謝秉昂

Reason :

number of ways that n + m open

parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses

(17)

### (Perrin Sequence)

B05203047 資工四 徐衍新

This sequence is interesting since p|P(p) where P(p) is the p-th member of the Perrin sequence and p is a prime number

Conversely, if n | P(n) , it doesn’t imply n is a prime number

But the counter example is rare, only two can be found below 10^6, that is, 271441 and 904631 I choose this sequencebecause I just found the sequence and prove p|P(p) independently, and thought that the counter statement may be true. After testing, I found 271441 and 904631 which

is the counter example below 10^6. By asking Google, I found that this is called “Perrin Pseudoprime” and this sequence has already been found. The smallest Perrin pseudoprime

271441 was found in 1982 by Adams and Shanks.

(18)

(19)

### Farey 序列

Farey sunburst of order 6

Comparison of Ford circles and a Farey diagram with circular arcs for n from 1 to 9. Each arc intersects its corresponding circles at right angles. In the SVG

image, hover over a circle or curve to highlight it and its terms.

R08946007

(20)

### [4, 9, 16, 25]

These are my student id since junior high school, they are all square number D08922025 曾奕青

(21)

### 2009]

Interesting? Why? KM Chao’s Life Big Events https://www.csie.ntu.edu.tw/~kmchao/basic.html

R08922054 資工碩一 張凱捷

(22)

### • These numbers are the height of top 20 peaks of Taiwan 100 Mountains( 百 岳 ). They show how high the back of Taiwan is, and demonstrate that why Taiwan is famous with high mountain.

• In addition, they also remind me the touched feeling when I stood on some of these peaks, especially the number 3952 which stands for Mt. Jade, the highest mountain in Taiwan.

• R08922079 洪浩翔

(23)

(24)

### Palindrome 迴文

1.

(25)

English Meaning:

From a distant sleep in the long night, everyone is awaken,

Is it the by the sound of the surfing boat?

Interesting Points:

Generally speaking it is quite difficult for Japanese to have long palindrome in the

“word” level due to the syntax

structure; nevertheless, this Japanese poem is delicately created with a artistic

concept. 25

なかきよの　とおのねふりの　みなめさめ なみのりふねの　おとのよきかな

### Palindrome

A palindrome is a word, number, phrase, or other sequence of

characters which reads the same backward as forward.

(26)

R07922138 傅家靖

(27)

Form wiki

### 璇璣圖

R07922181 黃子賢 作者是前秦女詩人蘇蕙。《璇璣

e.g. 黃書讀法

(28)
(29)

### HW1 Interesting Sequence

MACHI DIDI 2.0 ( 大人物 ) (feat. 熊仔 ) 最後一段歌詞 回文序列 :

============================

(30)

### 蘇軾 - 題金山寺

R08944005 網媒一 謝宏祺

(31)

### r08944028 網媒所 碩一 賴達

ere 是 before 的意思，原意為

“ 在看到 Elba 島之前，我無所不能” 另外有人翻譯成

“ 落敗孤島孤敗落”

Reference:

https://en.wikipedia.org/wiki/Palindrome

https://kknews.cc/zh-tw/history/4px3lp2.html

(32)

(33)

### 圖像詩

「是」，所有的「是」字組 成了「非」，有趣的地方在 於透過安排字的位置可以形 容一個成語。

「乒」或「乓」或

「丘」代表士兵的圖 像，「兵」是身體健 全的士兵，「乒」或

「乓」是斷一隻腳 的，「丘」是全斷腳 的，形容戰爭的過程 中不斷有士兵腳斷 掉，有趣在於用字的 圖像反映士兵真實的 樣態。

(34)

R08942125 廖克允

(35)

### Da Vinci Musical Notes in 'Last Supper'

I was deeply fascinated by this sequence of musical notes in my childhood.

The hand positions in the paint compose a sequence of musical notes. How

interesting it is!

R07921075 電機所 蕭大哲

The Last Supper is a late 15th-century mural painting by Italian artist Leonardo da Vinci

housed by the refectory of the Convent of Santa Maria delle Grazie in Milan, Italy. It is one of the Western world's most recognizable paintings.

An Italian musician has indicated that the

positions of hands and loaves of bread can be interpreted as notes on a musical staff and, if read from right to left, as was characteristic of Leonardo's writing, form a musical composition.

Source:

https://en.wikipedia.org/wiki/The_Last_Supper_(L eonardo)

https://stylesource01.wordpress.com/2007/11/14/h idden-music-in-last-supper-painting-da-vinci/

(36)

(37)

### In the beginning God created the heavens and the earth

(Genesis 1:1)

A Partial Listing of the Phenomenal Features of Sevens Found in Genesis 1:1 The number of Hebrew words = 7

The number of letters equals 28 (7x4 = 28)

The first 3 Hebrew words translated "In the beginning God created" have 14 letters (7x2 = 14)

The last four Hebrew words "the heavens and the earth" have 14 letters (7x2 = 14) The fourth and fifth words have 7 letters

The sixth and seventh words have 7 letters

The three key words: God, heaven and earth have 14 letters (7x2 = 14) The number of letters in the four remaining words is also 14 (7x2 = 14)

The middle word is the shortest with 2 letters. However, in combination with the word to the Right or left it totals 7 letters

The Hebrew numeric value of the first, middle and last letters is 133 (7x19 = 133) The Hebrew numeric value of first and last letters of all seven words is 1393

(7x199 = 1393) The hidden pattern of SEVENS in

the very first verse of the Bible.

Throughout the Bible the

number 7 appears repeatedly as a symbol of divine perfection, which makes me think this implies in the beginning God created the world with

perfection.

Ivan Panin carefully examined the Hebrew text of Genesis 1:1 and discovered an incredible phenomenon of multiples of 7 that could not be explained by chance. Genesis 1:1 was

composed of seven Hebrew words containing a total of 28 letters. In total, Panin

discovered 30 separate codes involving the number 7 in this first verse of the Bible.

P07922005 黃筑葭

(38)

Source: https://www.ettoday.net/news/20150524/510888.htm

Reason: 覺得很有趣，沒想到真的有這麼好笑的名字

R07922130

(39)

• 3 歲，不尿褲子

• 5 歲，能自己吃飯

• 18 歲，能自己開車

• 20 歲，有性生活

• 30 歲，有錢

• 40 歲，有錢

• 50 歲，有錢

• 60 歲，有性生活

• 70 歲，能自己開車

• 80 歲，能自己吃飯

• 90 歲，不尿褲子

(40)
(41)

r08922099 王 甯

Updating...

## References

Related subjects :