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A224165 Number of cycles the reduced complex Collatz function using 3 + (2n-1)i takes to hit 1 + i, or 0 if this never happens. 1
5, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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...

This is conjectured to be infinite and the same as in A224067.

For example, 3 + 41i yields -61 + 5i, -7 + -91i, 137 + -5i, 1 + 103i, -77 + 1i, -1 + -115i, 173 + -1i, 1 + 65i, -97 + 1i, -1 + -145i, 109 + -1i, 1 + 41i, -61 + 1i, -1 + -91i, 137 + -1i, 1 + 103i, -77 + 1i, -1 + -115i, 173 + -1i, 1 + 65i, -97 + 1i, -1 + -145i, 109 + -1i, 1 + 41i, which becomes cyclic about 1 + 41i.

LINKS

Table of n, a(n) for n=1..75.

Eric Weisstein, Mathworld: Collatz Problem

Wikipedia, Collatz Problem

EXAMPLE

a(5) = 13 because rcC(3 + 7i) yields -5 + 5i, -7 + -7i, 11 + -5i, 1 + 17i, -25 + 1i, -1 + -37i, 7 + -1i, 1 + 11i, -1 + 1i, -1 + -1i, 1 + -1i, 1 + 1i, which is 13 terms.

PROG

(JavaScript)

for (b=1; b<150; b+=2) {

c=1;

r[0]=3; 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. A224067.

Sequence in context: A262947 A265821 A266953 * A131286 A183390 A013607

Adjacent sequences:  A224162 A224163 A224164 * A224166 A224167 A224168

KEYWORD

nonn,less

AUTHOR

Jon Perry, Apr 02 2013

STATUS

approved

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Last modified November 17 23:26 EST 2019. Contains 329242 sequences. (Running on oeis4.)