A233399 A "turtle transform" of the primes p=A000040: coordinates (x,y)=(a(2n),a(2n+1)) of a walk on the complex plane, where at step n the cursor turns by arg(i^p(n)), then moves p(n) units ahead.

0, 0, -2, 0, -2, 3, -7, 3, -7, 10, 4, 10, 4, 23, -13, 23, -13, 42, 10, 42, 10, 71, 41, 71, 41, 108, 0, 108, 0, 151, 47, 151, 47, 204, 106, 204, 106, 265, 173, 265, 173, 194, 246, 194, 246, 115, 163, 115, 163, 26, 260, 26, 260, 127, 363, 127, 363, 20

Offset: 0

Author

M. F. Hasler, Dec 08 2013

Keywords

• sign
• walk

Examples

			The cursor starts for n=0 at the origin (a(0)=a(1)=0), facing East (= direction of the x-axis). At step 1, prime(1)=2 leads to a turn left by 2*90 degrees (now facing West), then a step of two units into the new direction, to x+iy = -2, thus a(2)=-2, a(3)=0. Prime(2)=3 leads to a turn left by 3*90 degrees, now facing North, and a step of 3 units in that direction, to x+iy = -2+3i, therefore a(4)=-2, a(5)=3.
		

Links

• M. F. Hasler, Turtle(prime) walk after 25 steps.
• M. F. Hasler, Turtle(prime) walk after 60 steps.
• M. F. Hasler, Turtle(prime) walk after the 99th step, drawn in red.

Programs

Mathematica

Turtle[v_] := Module[{p, L, d}, p=0; L={p}; d=1; For[i=1, i <= Length[v], i++, d *= I^v[[i]]; AppendTo[L, p += d*v[[i]] ] ]; L]; {Re[#], Im[#]}& /@ Turtle[Prime /@ Range[60]] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from PARI *)

PARI

Turtle(v,p=0,L=[p],d=1)=for(i=1,#v,d*=I^v[i];L=concat(L,p+=d*v[i]));L

PARI

a(nMax)=concat(apply(z->[real(z),imag(z)],Turtle(primes(nMax))))