# Transcendental-Natural Sequences

Definition: A sequence that has a transcendental number involved in the $n^{th}$ term and when $n$ is substituted and the result is rounded or left as it is, it produces all or most of the natural numbers.

Example:

$\frac{\pi}{4}$$+$ $\frac{x}{5}$$+$ $\frac{x^2}{69}$$+$ $\frac{x^3}{9168}$$+$ $\frac{x^4}{119999999}$

Result:

$1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 10, 10, 11, ...$

Challenge:

Prove that there is more sequences that meet the definition using algebraic proof or Python program.

Challenge $2$:

Prove that the sequence shows most of or all of the natural numbers using algebraic proof or Python program.

Note by Yajat Shamji
1 year ago

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All numbers don't exist in above sequence as

# Algebraic Proof

Above function is an increasing function, I.e. as we enter consecutive n's, the difference between consecutive values increase. At one point, the difference will go above 2 and one number won't exist.

# Code Proof

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 a = [] n = 1000000 #enter number of natural numbers here for i in range(1, n + 1): a.append(0) for i in range(1, n + 1): b = int(3.14/4 + i/5 + (i * i)/69 + (i*i*i)/9168 + (i*i*i*i)/119999999) if b > n: break a[b - 1] = 1 flag = 1 for i in range(1, n + 1): if a[i] == 0: print("All numbers don't exist as %s doesn't exists" % i) flag = 0 break if flag == 1: print("All numbers exist") 

 1 2 3 Output: All numbers don't exist as 18 doesn't exists 

@Yajat Shamji

- 1 year ago

Result?

- 1 year ago

Also, show the algebraic proof?

- 1 year ago

I showed.

- 1 year ago

For the second challenge, should I put doesn't? @Aryan Sanghi

And most in the definiton?

- 1 year ago

Yes, you should put.

- 1 year ago

Ok.

- 1 year ago

# Other sequences exist

A decreasing transcendental sequence can satisfy above equation as it is a decreasing function and difference will decrease between two consecutive numbers and will show all numbers eventually. @Yajat Shamji it's a challenge by me to find such function.

- 1 year ago

@Aryan Sanghi, I have made the note.

- 1 year ago

Are you sure that whenever n is substituted, the number is transcendental in above equation?

- 1 year ago

Yes. Should I add that? @Aryan Sanghi

- 1 year ago

No, then it is ok.

- 1 year ago

But what do you think, @Aryan Sanghi?

- 1 year ago

- 1 year ago

@Yajat Shamji I wasn't able to see your comment in the report. What did you say?

- 1 year ago

I deleted the report, that's why?

First comment was $11:30$am GMT?

Second comment was you really need proof (for the checkmate square), don't you?

- 1 year ago

I have class exactly at that time. What about 2:30?

- 1 year ago

GMT! Convert GMT to BST!

- 1 year ago

3:30 pm

- 1 year ago

$11:30$am GMT = $12:30$pm BST!

Seriously? You can't do that?

- 1 year ago

Oh. I thought it was +1 in net. It is +13, then maybe we can't play at such time differences.

- 1 year ago

In that case, I'll show you the checkmate square.

- 1 year ago