A265099 Least k such that floor(2^A006666(k)/3^A006667(k)) - k = n.

1, 6, 9, 19, 18, 27, 33, 37, 36, 50, 43, 56, 59, 66, 57, 74, 78, 72, 97, 87, 86, 98, 112, 119, 118, 134, 123, 115, 114, 130, 149, 148, 157, 135, 179, 144, 153, 187, 220, 174, 173, 172, 197, 196, 255, 224, 238, 219, 236, 203, 249, 268, 247, 246, 230, 229, 228

Offset

0,2

Comments

A006666 and A006667 are the number of halving and tripling steps to reach 1 in 3x+1 problem.

Conjecture: k exists for all n.

In other words, given an integer n, there always exists at least an integer k and a pair of integers (a, b) such that n + k = 2^a/3^b where a is the number of halving steps to reach 1, and b is the number of tripling steps to reach 1, in the 3x+1 problem.

Links

Table of n, a(n) for n=0..56.

Example

a(0) = 1 because A006666(1) = 0 and A006667(1) = 0 => floor(2^0/3^0) - 1 = 1 - 1 = 0;
a(1) = 6 because A006666(6) = 6 and A006667(6) = 2 => floor(2^6/3^2) - 6 = floor(64/9) - 6 = 7 - 6 = 1.

Mathematica

lst={}; Do[Collatz[k_]:=NestWhileList[If[EvenQ[#], #/2, 3 #+1]&, k, #>1&]; nn=500; t={}; k=0; While[Length[t]<nn, k++; c=Collatz[k]; ev=Length[Select[c, EvenQ]]; od=Length[c]-ev-1; If[Floor[2^ev/3^od]-k==n, AppendTo[lst, k]; Break[]]], {n, 0, 60}]; lst

Crossrefs

Cf. A006666, A006667, A075680, A211981, A225089.

Sequence in context: A297623 A293283 A078415 * A023041 A118277 A103186

Adjacent sequences: A265096 A265097 A265098 * A265100 A265101 A265102

Keyword

nonn

Author

Michel Lagneau, Dec 01 2015

Status

approved