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A063574
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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.
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4
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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
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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8 -> 12 -> 18 -> 27 -> 41 takes 4 steps so a(8) = 4.
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MATHEMATICA
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Table[Length[NestWhileList[If[EvenQ[#], (3#)/2, (3#+1)/2]&, n, Mod[#, 4]!= 1&]]-1, {n, 110}] (* Harvey P. Dale, Jul 06 2011 *)
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PROG
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(PARI) {stop=1000; for(n=1, 105, c=0; k=n; while((k%4)!=1&&c<stop, k=if(k%2==0, 3*k/2, (3*k+1)/2); c++); print1(if(c<stop, c, -1), ", "))}
(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
(Haskell)
a063574 n = fst $ until ((== 1) . flip mod 4 . snd)
(\(u, v) -> (u + 1, a007494 v)) (0, n)
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CROSSREFS
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KEYWORD
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easy,nice,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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