A133501 Number of steps for "powertrain" operation to converge when started at n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 5, 2, 3, 3, 1, 1, 1, 3, 2, 5, 5, 5, 4, 9, 1, 1, 2, 5, 3, 3, 4, 6, 3, 5, 1, 1, 3, 2, 3, 5, 3, 3, 2, 4, 1, 1, 6, 3, 4, 4, 3, 3, 8, 2, 1, 1, 6, 6, 2, 2, 3, 5, 3, 2, 1, 1, 5, 3, 4, 4, 5, 4, 3, 7, 1, 1, 2, 5, 4, 2, 3, 3, 2, 4, 1, 1, 1, 1, 1

Offset: 0

Author

J. H. Conway and N. J. A. Sloane, Dec 03 2007

Keywords

• nonn
• base

Comments

See A133500 for definition.

It is conjectured that every number converges to a fixed-point.

Examples

			39 -> 19683 -> 1594323 -> 38443359375 -> 59440669655040 -> 0, so a(39) = 5.
		

Links

• N. J. A. Sloane, Table of n, a(n) for n = 0..10000
• N. J. A. Sloane, Full trajectories of numbers from 1 to 10000

Crossrefs

For the powertrain map itself, see A133500.

See A133508, A133503 for records. See A135381 for high-water marks.

Programs

Maple

powertrain:=proc(n) local a,i,n1,n2,t1,t2; n1:=abs(n); n2:=sign(n); t1:=convert(n1, base, 10); t2:=nops(t1); a:=1; for i from 0 to floor(t2/2)-1 do a := a*t1[t2-2*i]^t1[t2-2*i-1]; od: if t2 mod 2 = 1 then a:=a*t1[1]; fi; RETURN(n2*a); end;
# Compute trajectory of n under repeated application of the powertrain map of A133500. This will return -1 if the trajectory does not converge to a single number in 100 steps (so it could fail if the trajectory enters a nontrivial loop or takes longer than 100 steps to converge).
PTtrajectory := proc(n) local p,M,t1,t2,i; M:=100; p:=[n]; t1:=n; for i from 1 to M do t2:=powertrain(t1); if t2 = t1 then RETURN(n,i-1,p); fi; t1:=t2; p:=[op(p),t2]; od; RETURN(n,-1,p); end;