cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Kevin Ge

Kevin Ge's wiki page.

Kevin Ge has authored 3 sequences.

A377945 Numbers k such that the trajectory of k under the `3x-1' map reaches k+1.

Original entry on oeis.org

1, 3, 9, 13, 15, 18, 19, 33, 49, 67, 73, 90, 109, 163, 391, 522, 607, 729, 810, 1093, 1639
Offset: 1

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Author

Kevin Ge, Nov 11 2024

Keywords

Comments

The 3x-1 map is x -> 3x-1 if x odd, and x -> x/2 if x even (A001281).
There are no more terms < 10^9.

Examples

			13 is a term because its trajectory 13 -> 38 -> 19 -> 56 -> 28 -> 14 -> ... reaches 13 + 1 = 14.

Crossrefs

Cf. A001281.

Programs

  • Python
    def isok(n):
    temp, loops = n, 0
    while(temp != n + 1 and loops<2):
        temp = temp // 2 if temp % 2 == 0 else 3 * temp - 1
        if(temp == 1 or temp == 5 or temp == 17):
            loops += 1
    return temp == n + 1
print([n for n in range(1, 2000) if isok(n)])

A377524 Number of steps for n to reach the minimum of its final cycle under iterations of the map (A123684): x->(3x-1)/2 if x odd, x/2 otherwise; or -1 if this never happens.

Original entry on oeis.org

0, 1, 3, 2, 0, 4, 2, 3, 7, 1, 5, 5, 6, 3, 7, 4, 0, 8, 5, 2, 5, 6, 2, 6, 10, 7, 4, 4, 8, 8, 4, 5, 12, 1, 9, 9, 9, 6, 10, 3, 6, 6, 7, 7, 14, 3, 11, 7, 11, 11, 8, 8, 12, 5, 8, 5, 20, 9, 9, 9, 5, 5, 13, 6, 25, 13, 13, 2, 14, 10, 14, 10, 10, 10, 7, 7, 11, 11, 11, 4
Offset: 1

Views

Author

Kevin Ge, Oct 28 2024

Keywords

Comments

The currently known cycle minimums are 1, 5, 17 and there are no known a(n) = -1 (trajectory never reaches a cycle).
This sequence is one way to extend A006666 (number of Collatz (3x+1)/2 steps) to the negative numbers.

Examples

			For n = 5, a(5) = 0 because 5 is already the minimum of its "final cycle".
For n = 12, a(12) = 6 because 12 takes 6 iterations to reach the minimum of its "final cycle": 12 -> 6 -> 3 -> 8 -> 4 -> 2 -> 1.

Crossrefs

Cf. A123684 ((3x-1)/2 map), A135730 (all steps).
Cf. A006666 (for (3x+1)/2).

Programs

  • Julia
    function three_x_minus_one_delay(n::Int)
    count = 0
    while (n != 1 && n != 5 && n != 17)
        if (isodd(n))
            n += n << 1 - 1
        end
        n >>= 1
        count += 1
    end
    return count
end

A368812 Numbers m such that Sum_{k=0..m-1} exp(2*Pi*i*k^5/m) != 0.

Original entry on oeis.org

1, 4, 8, 9, 11, 16, 25, 27, 31, 32, 36, 41, 44, 49, 61, 71, 72, 81, 88, 99, 100, 101, 108, 121, 124, 125, 128, 131, 144, 151, 164, 169, 176, 181, 191, 196, 200, 211, 216, 225, 241, 243, 244, 248, 251, 256, 271, 275, 279, 281, 284, 288, 289, 297, 311, 324, 328, 331, 341, 343, 352, 361, 369, 392, 396, 400, 401, 404
Offset: 1

Views

Author

Kevin Ge, Jan 06 2024

Keywords

Comments

Connect lines between the consecutive partial sums of Sum_{k=0..m-1} exp(2*Pi*i*k^5/m) != 0; this sequence gives values of m for which the resulting graph is "infinite."
A368959 is the intersection of all such sequences over exp(2*Pi*i*k^s/m), where s >= 2. Especially, all terms from A368959 are also here. - Vaclav Kotesovec, Jan 10 2024

Examples

			4 is a term because Sum_{k=0..3} exp(2*Pi*i*k^5/4) = 2 != 0.
11 is a term because Sum_{k=0..10} exp(2*Pi*i*k^5/11) = 1 + 10*cos(2*Pi/11) != 0.
12 is not a term because Sum_{k=0..11} exp(2*Pi*i*k^5/12) = 0.

Crossrefs

Cf. A001074, A042965 (Sum_{k=0..m-1} exp(2*Pi*i*k^(2n)/m) != 0 for all n>0).
Cf. A368959.

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