A257480 S(n) = (3 + (3/2)^v(1 + F(4*n - 3))*(1 + F(4*n - 3)))/6, n >= 1, where F(x) = (3*x + 1)/2^v(3*x + 1) for x odd, and v(y) denotes the 2-adic valuation of y.

1, 1, 5, 2, 4, 1, 8, 5, 7, 5, 41, 5, 10, 2, 17, 14, 13, 4, 32, 8, 16, 1, 26, 14, 19, 8, 68, 11, 22, 5, 35, 41, 25, 7, 59, 14, 28, 5, 44, 23, 31, 41, 365, 17, 34, 5, 53, 41, 37, 10, 86, 20, 40, 2, 62, 32, 43, 17, 149

Offset: 1

Author

L. Edson Jeffery, Apr 26 2015

Keywords

• nonn

Comments

In the following, let F^(k)(x) denote k-fold iteration of F and defined by the recurrence F^(k)(x) = F(F^(k-1)(x)), k > 0, with initial condition F^(0)(x) = x, and let S^(k)(n) denote k-fold iteration of S and defined by the recurrence S^(k)(n) = S(S^(k-1)(n)), k > 0, with initial condition S^(0)(n) = n, where F and S are as defined above.

Theorem 1: For each x, there exists a j>0 such that F^(j)(x) == 1 (mod 4).

Theorem 2: S(n) = m if and only if S(4*n-2) = m.

Conjecture 1: For each n, there exists a k such that S^(k)(n) = 1.

Theorem 3: Conjecture 1 is equivalent to the 3x+1 conjecture.

Theorem 4: The sequence {log(S(n))/log(n)}_{n>1} is bounded with least upper bound equal to log(3)/log(2).

[I have proved Theorems 1--4 (along with several lemmas) and am trying to finish typesetting the draft containing the proofs but had been too ill to finish that work until now. The draft also contains the derivation of the function S from properties of the known function F (A075677). When that paper is completed (hopefully within two weeks) I will then upload it to the links section and delete this comment.]

References

• K. H. Metzger, Untersuchungen zum (3n+1)-Algorithmus, Teil II: Die Konstruktion des Zahlenbaums, PM (Praxis der Mathematik in der Schule) 42, 2000, 27-32.

Links

• I. Korec and Štefan Znám, A Note on the 3x+1 Problem, Amer. Math. Monthly 94, 1987, pp. 771-772.
• J. C. Lagarias, The 3x + 1 Problem and Its Generalizations, Amer. Math. Monthly 92, 1985, pp. 3-23.
• J. C. Lagarias, The 3x+1 Problem: An Annotated Bibliography (1963-2000), arXiv:math/0309224 [math.NT], 2003-2011.
• J. C. Lagarias, The 3x+1 Problem: an annotated bibliography, II (2000-2009), arXiv:math/0608208 [math.NT], 2006-2012.

Crossrefs

Cf. A006370, A070165, A075677, A256598.

Cf. A241957, A254067, A254311, A257499, A257791 (all used in the proof of Thm 4).

Cf. A253676 (iteration of S terminating at the first occurrence of 1, assuming the 3x+1 conjecture).

Cf. A253720, A254068, A254070, A254131, A254312, A255138, A255168, A258415.

Programs

Mathematica

v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; s[n_] := (3 + (3/2)^v[1 + f[4*n - 3]]*(1 + f[4*n - 3]))/6; Table[s[n], {n, 59}]

PARI

a(n) = my(x=3*n-2, v=valuation(x, 2)); x>>=v; v=valuation(x+1, 2); (((x>>v)+1)*3^(v-1)+1)/2; \\ Ruud H.G. van Tol, Jul 30 2023