A224067 Number of cycles the reduced complex Collatz function using 1 + (2n-1)i takes to hit 1 + i, or 0 if this never happens.

1, 5, 5, 13, 9, 5, 13, 5, 9, 5, 17, 9, 13, 13, 5, 9, 21, 9, 9, 17, 0, 5, 9, 5, 13, 13, 13, 13, 17, 5, 9, 9, 0, 9, 9, 13, 13, 5, 17, 17, 17, 17, 17, 17, 21, 9, 5, 5, 25, 13, 5, 0, 25, 13, 13, 17, 9, 5, 13, 9, 33, 9, 9, 0, 0, 21, 9, 33, 21, 9, 13, 5, 13, 9, 17

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

Cf. A224165.

Sequence in context: A339947 A206553 A122213 * A049735 A055526 A146984

Adjacent sequences: A224064 A224065 A224066 * A224068 A224069 A224070

Keyword

nonn,less

Author

Jon Perry, Apr 02 2013

Status

approved