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

%I A324036 #49 Aug 10 2023 12:24:23
%S A324036 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,
%T A324036 77,13,83,43,89,15,95,49,101,17,107,55,113,19,119,61,125,21,131,67,
%U A324036 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
%N A324036 Modified reduced Collatz map fs acting on positive odd integers.
%C A324036 This is a modification of the reduced Collatz map given in A075677.
%C A324036 The Collatz conjecture is that iteration of the map fs leads to 1 for all positive odd integers.
%C A324036 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.
%C A324036 The advantage of the map fs (or a) over fCr (or b) is an explicit formula over a recurrence.
%C A324036 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.
%C A324036 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).
%C A324036 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.
%C A324036 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).
%H A324036 Wolfdieter Lang, <a href="/A324036/a324036.pdf">Collatz Trees from Vaillant-Delarue Maps</a>
%H A324036 Nicolas Vaillant and Philippe Delarue, <a href="https://web.archive.org/web/20220317020641/http://nini-software.fr/site/uploads/arithmetics/collatz/Intrinsic%203x+1%20V2.01.pdf">The hidden face of the 3x+1 problem. Part I: Intrinsic algorithm</a>, April 26 2019.
%F A324036 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.
%F A324036 Conjectures from _Colin Barker_, Oct 14 2019: (Start)
%F A324036 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).
%F A324036 a(n) = 2*a(n-4) - a(n-8) for n>7.
%F A324036 (End)
%e A324036 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.
%e A324036 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.
%e A324036 The first additional N = 3 steps occur for 213 = a(106): 213, 53, 13, 3, 5, 1 versus 213, 5, 1 for fCr.
%e A324036 The first additional N = 4 steps occur for 853 = a(426): 853, 213, 53, 13, 3, 5, 1 versus 853, 5, 1 for fCr.
%o A324036 (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
%Y A324036 Cf. A000302, A072197, A075677, A324037, A324038.
%K A324036 nonn,easy
%O A324036 0,2
%A A324036 _Nicolas Vaillant_, Philippe Delarue, _Wolfdieter Lang_, May 08 2019
%E A324036 More terms from _Michel Marcus_, Aug 10 2023