A287798 Least k such that A006667(k)/A006577(k) = 1/n.

159, 6, 5, 10, 20, 40, 80, 160, 320, 640, 1280, 2560, 5120, 10240, 20480, 40960, 81920, 163840, 327680, 655360, 1310720, 2621440, 5242880, 10485760, 20971520, 41943040, 83886080, 167772160, 335544320, 671088640, 1342177280, 2684354560, 5368709120, 10737418240

Offset: 3

Author

Michel Lagneau, Jun 01 2017

Keywords

• nonn
• easy

Comments

A006667: number of tripling steps to reach 1 in '3x+1' problem.

A006577: number of halving and tripling steps to reach 1 in '3x+1' problem.

a(n) = {159, 6} union {A020714}.

Examples

			a(3) = 159 because A006667(159)/A006577(159) = 18/54 = 1/3.
		

Links

• Colin Barker, Table of n, a(n) for n = 3..1000
• Index entries for linear recurrences with constant coefficients, signature (2).

Crossrefs

Cf. A006577, A006666, A006667. Essentially the same as A020714, A084215, A146523 and A257113.

Programs

Maple

nn:=10^12:
for n from 3 to 35 do:
ii:=0:
for k from 2 to 10^6 while(ii=0) do:
  m:=k:s1:=0:s2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m,2)=0
     then
     s2:=s2+1:m:=m/2:
     else
     s1:=s1+1:m:=3*m+1:
    fi:
   od:
    if n*s1=s1+s2
     then
     ii:=1: printf(`%d, `,k):
     else
    fi:
od:od:

Mathematica

f[u_]:=Module[{a=u,k=0},While[a!=1,k++;If[EvenQ[a],a=a/2,a=a*3+1]];k];Table[f[u],{u,10^7}];g[v_]:=Count[Differences[NestWhileList[If[EvenQ[#],#/2,3#+1]&,v,#>1&]],_?Positive];Table[g[v],{v,10^7}];Do[k=3;While[g[k]/f[k]!=1/n,k++];Print[n," ",k],{n,3,35}]

PARI

a(n) = if(n < 5, [0,0,159,6][n], 5<<(n-5)) \\ David A. Corneth, Jun 01 2017

PARI

Vec(x^3*(159 - 312*x - 7*x^2) / (1 - 2*x) + O(x^50)) \\ Colin Barker, Jun 01 2017

Formula

For n >= 5, a(n) = 5*2^n/32. - David A. Corneth, Jun 01 2017

From Colin Barker, Jun 01 2017: (Start)

G.f.: x^3*(159 - 312*x - 7*x^2) / (1 - 2*x).

a(n) = 2*a(n-1) for n>5.

(End)