A064433 Number of iterations of A064455 to reach 2 (or 1 in the case of 1).

1, 1, 2, 6, 3, 5, 7, 12, 4, 14, 6, 11, 8, 8, 13, 13, 5, 10, 15, 15, 7, 7, 12, 12, 9, 17, 9, 71, 14, 14, 14, 68, 6, 19, 11, 11, 16, 16, 16, 24, 8, 70, 8, 21, 13, 13, 13, 67, 10, 18, 18, 18, 10, 10, 72, 72, 15, 23, 15, 23, 15, 15, 69, 69, 7, 20, 20, 20, 12, 12, 12, 66, 17, 74, 17

Offset: 1

Author

Jonathan Ayres (Jonathan.ayres(AT)btinternet.com), Oct 01 2001

Keywords

• nonn
• easy
• look

Comments

Similar to 3x+1 series (A008908). Does this sequence converge to 2 for all values of n (true for all values of n up to 100000)? The inverse sequence using next n = n-int(n/2) for n even and n+int(n/2) for n odd leads to 3 (?) possible end sequences (1), (5, 7, 10) and (17, 25, 37, 55, 82, 41, 61, 91, 136, 68, 34)

Starting with a number n, the next value generated is n+int(n/2) if n is even, n-int(n/2) if n is odd; a(n) is the number of iteration for the initial value n to reach the limit of 1 to 2

Collatz's 3N+1 function as isometry over the dyadics is N->N/2 if even, but N->(3N+1)/2 if odd, including the (necessary) halving into each tripling step. Counting steps until reaching 1 in this way leads to this sequence instead of A008908. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009

The value at each step of a trajectory starting with n (n>1) is equal to the value plus one at the same step of the row starting with (n-1) of the irregular triangle of the abbreviated (Terras-modified) Collatz sequence (A070168). - K. Spage, Aug 07 2014

Examples

			a(4) = 6. Starting with 4, 4 is even so the next number is 4+int(4/2) = 6, 6 is even so next number is 6+int(6/2) = 9, 9 is odd so next number is 9-int(9/2) = 5, 5 is odd so next number is 5-int(5/2) = 3, 3 is odd so next number is 3-int(3/2)=2, so giving a sequence of 4,6,9,5,3,2: 6 numbers.
a(5) = 3. Starting with 5, A064455(5) = 3, A064455(3) = 2, so giving a trajectory of 5,3,2: 3 numbers. - _K. Spage_, Aug 07 2014
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..19683
• M. del P. Canales Chacon and M. J. Vielhaber, Structural and Computational Complexity of Isometries and Their Shift Commutators, Electr. Colloq. on Computational Cpx., ECCC TR04-057, 2004. [From Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009]

Crossrefs

Cf. A006666, A008908, A064455, A070168.

Programs

Mathematica

Table[Length@ NestWhileList[If[EvenQ@ #, 3 #/2, (# + 1)/2] &, n, # != 1 + Boole[n > 1] &], {n, 75}] (* Michael De Vlieger, Sep 24 2016 *)

PARI

A064455(n) = {if(n%2, (n + 1)/2, 3*n/2)}
A064433(n) = {my(c=1); if(n==1, 1, while(n!=2, n=A064455(n); c++); c)} \\ K. Spage, Aug 07 2014

Formula

a(n) = A006666(n-1) + 1. - K. Spage, Aug 04 2014