Transcendental-Natural Sequences

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

Example:

π4\frac{\pi}{4}++ x5\frac{x}{5}++ x269\frac{x^2}{69}++ x39168\frac{x^3}{9168}++ x4119999999\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,...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 22:

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

Note by Yajat Shamji
1 week ago

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1 vote

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Comments

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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

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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")

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Output:

All numbers don't exist as 18 doesn't exists

@Yajat Shamji

Aryan Sanghi - 1 week ago

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Result?

Yajat Shamji - 1 week ago

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Also, show the algebraic proof?

Yajat Shamji - 1 week ago

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I showed.

Aryan Sanghi - 1 week ago

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

And most in the definiton?

Yajat Shamji - 1 week ago

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Yes, you should put.

Aryan Sanghi - 1 week ago

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@Aryan Sanghi Ok.

Yajat Shamji - 1 week ago

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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.

Aryan Sanghi - 1 week ago

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@Aryan Sanghi, I have made the note.

Yajat Shamji - 1 week ago

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

Aryan Sanghi - 1 week ago

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Yes. Should I add that? @Aryan Sanghi

Yajat Shamji - 1 week ago

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@Yajat Shamji No, then it is ok.

Aryan Sanghi - 1 week ago

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@Aryan Sanghi But what do you think, @Aryan Sanghi?

Yajat Shamji - 1 week ago

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

Aryan Sanghi - 1 week ago

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I deleted the report, that's why?

First comment was 11:3011:30am GMT?

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

Yajat Shamji - 1 week ago

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

Aryan Sanghi - 1 week ago

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@Aryan Sanghi GMT! Convert GMT to BST!

Yajat Shamji - 1 week ago

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@Yajat Shamji 3:30 pm

Aryan Sanghi - 1 week ago

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@Aryan Sanghi 11:3011:30am GMT = 12:3012:30pm BST!

Seriously? You can't do that?

Yajat Shamji - 1 week ago

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

Aryan Sanghi - 1 week ago

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

Yajat Shamji - 1 week ago

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