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.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Transcendental-Natural Sequences

Comments

Algebraic Proof

Code Proof

Other sequences exist