|
| |
|
|
A239126
|
|
Rectangular array showing the starting values M(n, k), k >= 1, for the Collatz operation (ud)^n, n >= 1, ending in an odd number, read by antidiagonals.
|
|
9
|
|
|
|
3, 7, 7, 11, 15, 15, 15, 23, 31, 31, 19, 31, 47, 63, 63, 23, 39, 63, 95, 127, 127, 27, 47, 79, 127, 191, 255, 255, 31, 55, 95, 159, 255, 383, 511, 511, 35, 63, 111, 191, 319, 511, 767, 1023, 1023, 39, 71, 127, 223, 383, 639, 1023, 1535, 2047, 2047
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,1
|
|
|
COMMENTS
|
The companion array and triangle for the odd end numbers N(n, k) is given in A239127.
The two operations on natural numbers m used in the Collatz 3x+1 conjecture are here (following the M. Trümper paper given in the link) denoted by u for 'up' and d for 'down': u m = 3*m+1, if m is odd, and d m = m/2 if m is even. The present array gives all start numbers M(n, k) for the Collatz word (ud)^n = s^n (s = ud is useful because, except for the one letter word u, at least one d follows a letter u), with n >= 1, and k >= 1. Such Collatz sequences have the maximal number of u's (grow fastest).
This rectangular array is M of Example 2.2. with x=y = n, n >= 1, of the M. Trümper reference, pp. 7-8, written as a triangle by taking NE-SW diagonals. The Collatz sequence starting with M(n, k) has length 2*n+1 for each k and it ends in the odd number N(n, k) given in A239127.
|
|
|
LINKS
|
|
|
|
FORMULA
|
The array: M(n, k) = 2^(n+1)*k - 1 for n >= 1 and k >= 1.
The triangle: TM(m, n) = M(n, m-n+1) = 2^(n+1)*(m-n+1) - 1 for m >= n >= 1 and 0 for m < n.
|
|
|
EXAMPLE
|
The rectangular array M(n, k) begins:
n\k 1 2 3 4 5 6 7 8 9 10 ...
1: 3 7 11 15 19 23 27 31 35 39
2: 7 15 23 31 39 47 55 63 71 79
3: 15 31 47 63 79 95 111 127 143 159
4: 31 63 95 127 159 191 223 255 287 319
5: 63 127 191 255 319 383 447 511 575 639
6: 127 255 383 511 639 767 895 1023 1151 1279
7: 255 511 767 1023 1279 1535 1791 2047 2303 2559
8: 511 1023 1535 2047 2559 3071 3583 4095 4607 5119
9: 1023 2047 3071 4095 5119 6143 7167 8191 9215 10239
10: 2047 4095 6143 8191 10239 12287 14335 16383 18431 20479
...
The triangle TM(m, n) begins (zeros are not shown):
m\n 1 2 3 4 5 6 7 8 9 10 ...
1: 3
2: 7 7
3: 11 15 15
4: 15 23 31 31
5: 19 31 47 63 63
6: 23 39 63 95 127 127
7: 27 47 79 127 191 255 255
8: 31 55 95 159 255 383 511 511
9: 35 63 111 191 319 511 767 1023 1023
10: 39 71 127 223 383 639 1023 1535 2047 2047
...
---------------------------------------------------------------------
n=1, ud, k=1: M(1, 1) = 3 = TM(1, 1), N(1,1) = 5 with the Collatz sequence [3, 10, 5] of length 3.
n=1, ud, k=2: M(1, 2) = 7 = TM(2, 1), N(1,2) = 11 with the Collatz sequence [7, 22, 11] of length 3.
n=4, (ud)^4, k=2: M(4, 2) = 63 = TM(5, 4), N(4,2) = 323 with the Collatz sequence [63, 190, 95, 286, 143, 430, 215, 646, 323] of length 9.
n=5, (ud)^5, k=1: M(5, 1) = 63 = TM(5, 5), N(5,1) = 485 with the Collatz sequence [63, 190, 95, 286, 143, 430, 215, 646, 323, 970, 485] of length 11.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
