A133508 -id:A133508 - cp's OEIS frontend

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

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

See A133500 for definition.

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

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

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

For the powertrain map itself, see A133500.

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

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;

A133503 Numbers for which iteration of the powertrain map of A133500 takes a record number of steps to converge.

Original entry on oeis.org

0, 10, 24, 26, 39, 3573, 26899, 68697, 497699, 3559595, 555959597395

Offset: 1

J. H. Conway and N. J. A. Sloane, Dec 03 2007, Dec 04 2007, Dec 18 2007

Where records occur in A133501.
This sequence is almost certainly finite.
The number 31395559595973 takes 16 steps to converge and may be the next term. It may also be the last term.
The next term is > 10^7 (and <= 31395559595973).

			The smallest number that takes 13 steps to converge is 497699, for which the trajectory is 497699 -> 11948427342082473984 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 15 steps to converge is 3559595 -> for which the trajectory is 3559595 -> 4634857177734375 -> 23122964691361341376561152 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The number 31395559595973 takes 16 steps to converge and so the next term is >= 16. The trajectory is 31395559595973 -> 471570692025125026702880859375 -> 34755118508614725279865110528 -> 23122964691361341376561152000000 -> 1194842734208247398400000000 -> 23554621393597287150649344 -> 2030652382202824185652602470400000 -> 101921587200000000 -> 38281250 -> 1679616 -> 1452729852 -> 1318305830625 -> 70312500 -> 96 -> 531441 -> 500 -> 0.
The smallest number that takes 16 steps to converge is 555959597395, for which the trajectory starts 555959597395 -> 471570692025125026702880859375 and then continues as above. - _Michael S. Branicky_, Jan 24 2022

See A133508 for the corresponding numbers of steps. Cf. A133500, A133501.

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

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