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 A265946 Sums of positive differences and negated sums of negative differences in the Collatz trajectory of n. 1
 0, 0, 0, 1, 18, 20, 0, 3, 11, 15, 18, 23, 111, 117, 0, 7, 130, 138, 11, 20, 96, 106, 18, 29, 38, 50, 111, 124, 267, 281, 0, 15, 73, 89, 130, 147, 194, 212, 11, 30, 43, 63, 96, 117, 236, 258, 18, 41, 245, 269, 38, 63, 40563, 40589, 111, 138, 155, 183, 267, 296 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS For each number n, we consider the Collatz trajectory represented by the ordered sequence {x(i)}, i = 1, 2, ..., r starting with x(1) = n and ending with x(r) = 1. Let d(i) be the difference x(i+1) - x(i). This sequence lists the pairs of positive integers (u, -v) such that u is the sum of all the positive differences and v is the sum of all the negative differences. I present an interesting conjecture because of the linearity of the result. Conjecture: a(2n) - a(2n-1) = n - 1. Another definition: Consider F(x) the 3x + 1 function iterates (x, T(x), T(T(x)), ..., 4, 2, 1) plotted on standard vertical and horizontal scale where each point (0, x), (1, T(x)), (2, T(T(x)), ..., (r, 1) is connected to the next by a straight line with r being the number of iterations needed to reach 1. Let F'(x) be the derivative of F(x). The sequence lists the pairs of integers (u, -v) such that u is the sum of the positive values of F'(x) and v is the sum of the negative values of F'(x) at x = 1, 2, ..., r. Property: a(2^n - 1) = 0. In the general case, if m is an odd integer, a(m) = q => a(m*2^r + 2^r - 1) = q, r = 0,1,2,.. Example: for m = 5, a(5)=18 => a(11) = a(23) = a(47) = a(95) = ... = 18. LINKS Michel Lagneau, Table of n, a(n) for n = 1..10000 EXAMPLE n=3 => (a(2n-1), a(2n)) = (a(5),a(6)) = (18,20) because the Collatz trajectory of 3 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. The sum of the positive differences is u = (10 - 3) + (16 - 5) = 18 and the sum of the negative differences is u = (5 - 10) + (8 - 16) + (4 - 8) + (2 - 4) + (1 - 2)= -20 with the property a(6) - a(5) = 3 - 1 = 2. MAPLE T:=array(1..1000):nn:=1000: for n from 1 to 100 do:   kk:=1:m:=n:T[kk]:=n:it:=0:     for i from 1 to nn while(m<>1) do:      if irem(m, 2)=0       then       m:=m/2:kk:=kk+1:T[kk]:=m:it:=it+1:       else       m:=3*m+1:kk:=kk+1:T[kk]:=m:it:=it+1:      fi:     od:    jj:=0:s1:=0:s2:=0:     for j from 1 to it do:      if T[j]

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Last modified September 27 15:28 EDT 2021. Contains 347691 sequences. (Running on oeis4.)