OFFSET
1,2
COMMENTS
The complex Collatz function takes a complex number z to 3iz + (1+i). The resulting real part is divided by 2 until it's odd, and the same for the imaginary part.
The effect here is that say 4 + 14i is reduced to 1 + 7i.
The reduced complex Collatz function does all this in the same cycle.
Equals 0 for n = 21, 33, 52, 64, 65, 81, 82, 101, 103, 127, 129, 130, 163, 201, 204, 206, 253, 254, 256, 258, 259, 313...
This is conjectured to be infinite and the same as in A224165.
For example, 1 + 41i yields -61 + 1i, -1 + -91i, 137 + -1i, 1 + 103i, -77 + 1i, -1 + -115i, 173 + -1i, 1 + 65i, -97 + 1i, -1 + -145i, 109 + -1i, 1 + 41i - cyclic period 12.
From Robert G. Wilson v, Apr 04 2013: (Start)
If n>0 a(n) is odd, if n<1 a(n) is even. In fact, for n>0, a(n) is of the form 4k-3 and for n<1, a(n) is of the form 4k-2.
For a(k)=0 for k: 21, 33, 52, 64, 65, 81, 82, 101, 103, 127, 129, 130, 163, 201, 204, 206, 253, 254, 256, 258, 259, 313, 317, 320, 322, 324, 326, 396, 400, 401, 402, 404, 405, 407, 408, 409, 410, 501, 505, 506, 508, 511, 512, 514, 515, 518, 625, 635, 639, 641, 643, 645, 647, 648, 650, 791, 799, 800, 802, 803, 807, 811, 812, 814, 815, 816, 819, 822, 988, 989, 1003, ... .
Amongst the first 100000 positive terms, 6.726% are 0.
The first occurrence of 4k-3 occurs at: 1, 2, 5, 4, 11, 17, 49, 77, 61, 96, 76, 476, 377, 301, 509, 804, 587, 941, 1585, 1669, 1348, 533, 1683, 333, 132, 208, 656, 260, 820, 2585, 4091, 3229, 5101, 8068, 6381, 1261, 3980, 3148, 2485, 15684, 12409, 4907, 15473, 6125, 9681, 3825, 6044, 19100, 60235, 23797, 37605, 59425, 23477, 37099, 29313, 23161, ... .
The first negative terms beginning at 0: {2, 6, 2, 10, 6, 14, 6, 10, 6, 6, 2, 14, 6, 10, 10, 14, 6, 6, 6, 22, 10, 14, 14, 14, 14, 6, 6, 10, 0, 10, 10, 6, 14, 6, 6, 0, 18, 18, 6, 10, 10, 10, 2, 14, 10, 0, 14, 14, 14, 14, 6, 18, 10, 34, 10, 10, 6, 0, 10, 34, 22, 10, 14, 14, 14, 14, 6, 18, 18, 18, 6, ... .
For a(-k)=0 for k: {28, 35, 45, 57, 72, 112, 114, 142, 143, 144, 145, 175, 179, 180, 182, 224, 225, 228, 229, 230, 277, 283, 285, 287, 288, 289, 290, 356, 361, 363, 444, 448, 449, 450, 453, 456, 457, 458, 563, 567, 570, 572, 574, 575, 577, 578, 582, 702, 709, 711, 712, 713, 716, 718, 722, 724, 727, 728, 730, 877, 889, 896, 897, 898, 900, 901, 902, 907, 908, 909, 911, 912, 913, 914, 916, 918, 919, 920, 922, 1110, ... .
Amongst the first 100000 negative terms, 7.256% are 0.
The first occurrence of 4k-2, beginning with n=0, occurs at: 0, 1, 3, 5, 36, 19, 95, 75, 53, 84, 133, 211, 668, 571, 451, 659, 1043, 892, 1876, 1515, 1171, 1892, 747, 147, 116, 368, 291, 460, 1453, 2300, 3635, 2869, 4533, 14341, 2835, 2240, 3540, 2797, 4416, 6979, 5519, 17427, 6891, 10887, 8604, 1699, 5371, 33955, 53541, 42304, 66851, 52821, 41735, 65952, 13027, 82347, ... .
(End)
LINKS
Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Eric Weisstein, Mathworld: Collatz Problem
Wikipedia, Collatz Problem
EXAMPLE
a(4) = 13 because rcC(1 + 7i) yields 1 + 7i, -5 + 1i, -1 + -7i, 11 + -1i, 1 + 17i, -25 + 1i, -1 + -37i, 7 + -1i, 1 + 11i, -1 + 1i, -1 + -1i, 1 + -1i, 1 + 1i which is 13 terms.
MATHEMATICA
g[z_] := Block[{x = 3I*z + (1 + I)}, While[ EvenQ[ Re[x]], x = Re[x]/2 + Im[x]I ]; While[ EvenQ[ Im[x]], x = Re[x] + Im[x]/2*I]; x]; f[n_] := f[n] = Block[{k = 1, z = 1 + (2n - 1)I}, While[z != (1 + I) && k < 1000001, z = g[z]; k++]; If[k == 1000001, k = 0, k]]; Array[f, 70] (* Robert G. Wilson v, Apr 04 2013 *)
PROG
(JavaScript)
for (b=1; b<150; b+=2) {
c=1;
r[0]=1; i[0]=b;
while (r[c-1]!=1 || i[c-1]!=1 && c<1000) {
i[c]=r[c-1]*3+1;
r[c]=-i[c-1]*3+1;
while (r[c]%2==0) r[c]/=2;
while (i[c]%2==0) i[c]/=2;
c++;
}
document.write(c+", ");
}
CROSSREFS
KEYWORD
nonn,less
AUTHOR
Jon Perry, Apr 02 2013
STATUS
approved