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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.

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A324036 Modified reduced Collatz map fs acting on positive odd integers.

Original entry on oeis.org

1, 5, 1, 11, 7, 17, 3, 23, 13, 29, 5, 35, 19, 41, 7, 47, 25, 53, 9, 59, 31, 65, 11, 71, 37, 77, 13, 83, 43, 89, 15, 95, 49, 101, 17, 107, 55, 113, 19, 119, 61, 125, 21, 131, 67, 137, 23, 143, 73, 149, 25, 155, 79, 161, 27, 167, 85, 173, 29, 179, 91, 185, 31, 191, 97, 197, 33, 203, 103, 209
Offset: 0

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Author

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 08 2019

Keywords

Comments

This is a modification of the reduced Collatz map given in A075677.
The Collatz conjecture is that iteration of the map fs leads to 1 for all positive odd integers.
In the Vaillant-Delarue (V-D) reference the present map fs: Odd -> Odd, 2*n+1 -> a(n) = fs(2*n+1), for n >= 0, is called f_{s}. The differences from b(n) = A075677(n+1) = fCr(2*n+1) (called f_{cr} in V-D) occur for the positions n = 2 + 4*k, for k >= 1: b(2 + 4*k) = b(k) = A075677(k+1) but a(2 + 4*k) = 1 + 2*k, which differs.
The advantage of the map fs (or a) over fCr (or b) is an explicit formula over a recurrence.
Additional steps are introduced in the iteration of fs versus fCr. This leads to an incomplete binary tree, called CfsTree, given in A324038. No such tree is available for fCr.
Such additional steps in fs can only occur after odd numbers congruent to 5 modulo 8: fs(5 + 8*k) = a(2 + 4*k) = 1 + 2*k and fs(1 + 2*k) = a(k). On the other hand, fCr(5 + 8*k) = b(2 + 4*k) = b(k).
The appearance of exactly N consecutive steps in fs versus fCr, for N >= 2, can be shown recursively to start with the odd numbers O(N;k) = 1 + 4*O(N-1;k), for N >= 3, with input O(2;k) = 53 + (4^3)*k. These are the numbers O(N;k) = A072197(N) + A000302(N+1)*k, for N >= 2. Therefore only one additional step follows directly after an odd number 5 (mod 8) if it is not of the O(N;k) type for N >= 2.
The minimal number of iterations of function fs acting on 2*n + 1 (or a acting on n), for n >= 0, to reach 1 is given in A324037 (if for very large n the number 1 should not be reached A324037(n) is set to -1).

Examples

			Iteration of fs on 11: 11, 17, 13, 3, 5, 1, whereas for fCr: 11, 17, 13 , 5, 1. The additional step (N = 1) occurs for 13 == 5 (mod 8), and 13 does not belong to the O(N;k) sets for N >= 2.
The first additional N = 2 steps occur for 53 = a(26): 53, 13, 3, 5, 1, versus iteration of fCr: 53, 5, 1. Such N = 2 steps occur precisely after 53 + 64*k as 13 + 16*k and 3 + 4*k.
The first additional N = 3 steps occur for 213 = a(106): 213, 53, 13, 3, 5, 1 versus 213, 5, 1 for fCr.
The first additional N = 4 steps occur for 853 = a(426): 853, 213, 53, 13, 3, 5, 1 versus 853, 5, 1 for fCr.

Links

Crossrefs

Cf. A000302, A072197, A075677, A324037, A324038.

Programs

  • PARI
    a(n) = my(m=Mod(n,4)); if (m==0, (2 + 3*n)/2, if (m==2, n/2, 2 + 3*n)); \\ Michel Marcus, Aug 10 2023

Formula

a(n) = fs(1 + 2*n) = (2 + 3*n)/2 if n == 0 (mod 4), a(n) = 2 + 3*n, for n == 1 or 3 (mod 4), and a(n) = n/2 if n == 2 (mod 4). This corresponds to fs(1 + 8*k) = 1 + 6*k, fs(3 + 8*k) = 5 + 12*k, fs(5 + 8*k) = 1 + 2*k, and fs(7 + 8*k) = 11 + 12*k, for k >= 0.
Conjectures from Colin Barker, Oct 14 2019: (Start)
G.f.: (1 + 5*x + x^2 + 11*x^3 + 5*x^4 + 7*x^5 + x^6 + x^7) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2).
a(n) = 2*a(n-4) - a(n-8) for n>7.
(End)

Extensions

More terms from Michel Marcus, Aug 10 2023

A324245 The modified Collatz map considered by Vaillant and Delarue.

Original entry on oeis.org

0, 2, 0, 5, 3, 8, 1, 11, 6, 14, 2, 17, 9, 20, 3, 23, 12, 26, 4, 29, 15, 32, 5, 35, 18, 38, 6, 41, 21, 44, 7, 47, 24, 50, 8, 53, 27, 56, 9, 59, 30, 62, 10, 65, 33, 68, 11, 71, 36, 74, 12, 77, 39, 80, 13, 83, 42, 86, 14, 89, 45, 92, 15, 95, 48, 98, 16, 101, 51, 104, 17, 107, 54, 110, 18, 113, 57, 116, 19, 119, 60
Offset: 0

Views

Author

Nicolas Vaillant, Philippe Delarue, Wolfdieter Lang, May 09 2019

Keywords

Comments

This is a modified Collatz-Terras map (A060322), called in the Vaillant and Delarue link f.
The Collatz conjecture: iterations of the map f = a: N_0 -> N_0 with n -> a(n) lead always to 0.
The minimal number k with a^{[k]}(n) = 0 is given by A324037(n).
The tree CfTree, related to this map, giving the branches which lead to 0 for each vertex label of level n >= 0 is given in A324246.

Links

Crossrefs

Cf. A060322, A324037, A324246, A349414.

Programs

  • Mathematica
    a[n_]:=If[OddQ@n,(3n+1)/2,If[Mod[n,4]==0,3n/4,(n-2)/4]];Array[a,51,0] (* Giorgos Kalogeropoulos, Dec 08 2021 *)
  • PARI
    A324245(n) = if(n%2, (1+3*n)/2, if(!(n%4), 3*(n/4), (n-2)/4)); \\ (After Mathematica-code) - Antti Karttunen, Dec 09 2021

Formula

a(n) = (3*n+1)/2 if n is odd, 3*n/4 if n == 0 (mod 4), and (n-2)/4 if n == 2 (mod 4).
a(n) = A349414(n) + n. - Ruud H.G. van Tol, Dec 08 2021
G.f.: x*(2 + 5*x^2 + 3*x^3 + 4*x^4 + x^5 + x^6)/(1 - x^4)^2. - Stefano Spezia, Dec 08 2021

Extensions

More terms from Antti Karttunen, Dec 09 2021
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