A063574 Number of steps to reach an integer == 1 (mod 4) when iterating the map n -> 3n/2 if n even or (3n+1)/2 if n odd.

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

Offset: 1

Author

N. J. A. Sloane, Sep 23 2002

Keywords

• easy
• nice
• nonn

Examples

			8 -> 12 -> 18 -> 27 -> 41 takes 4 steps so a(8) = 4.
		

References

• L. Flatto, Z-numbers and beta-transformations, in Symbolic dynamics and its applications (New Haven, CT, 1991), 181-201, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (1968), pp. 313-321.

Crossrefs

Cf. A007494.

Programs

Haskell

a063574 n = fst $ until ((== 1) . flip mod 4 . snd)
                        (\(u, v) -> (u + 1, a007494 v)) (0, n)
-- Reinhard Zumkeller, Dec 13 2014

Mathematica

Table[Length[NestWhileList[If[EvenQ[#],(3#)/2,(3#+1)/2]&,n, Mod[#,4]!= 1&]]-1,{n,110}] (* Harvey P. Dale, Jul 06 2011 *)

PARI

{stop=1000; for(n=1,105,c=0; k=n; while((k%4)!=1&&c
				

PARI

b(n)=valuation(n,2); a(n)=b(n)+b((3^b(n)*n/2^b(n)+1)/2) - Lambert Herrgesell (zero815(AT)googlemail.com) and Lambert Klasen (lambert.klasen(AT)gmx.net), Apr 24 2006

Formula

For odd n: a(n)=A007814(n+1), for even n: A007814(n) steps until an odd number is reached, which leads directly to the formula: with b(n)=A007814(n) (binary carry sequence) a(n)=b(n)+b((3^b(n)*n/2^b(n)+1)/2) - Lambert Herrgesell (zero815(AT)googlemail.com) and Lambert Klasen (lambert.klasen(AT)gmx.net), Apr 24 2006. Hence in particular, a(n) is well-defined.

Extensions

Extended by Klaus Brockhaus, Sep 23 2002