cp's OEIS Frontend

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.

%I A257480 #51 Oct 20 2023 10:26:53
%S A257480 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,
%T A257480 22,5,35,41,25,7,59,14,28,5,44,23,31,41,365,17,34,5,53,41,37,10,86,20,
%U A257480 40,2,62,32,43,17,149
%N 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.
%C A257480 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.
%C A257480 Theorem 1: For each x, there exists a j>0 such that F^(j)(x) == 1 (mod 4).
%C A257480 Theorem 2: S(n) = m if and only if S(4*n-2) = m.
%C A257480 Conjecture 1: For each n, there exists a k such that S^(k)(n) = 1.
%C A257480 Theorem 3: Conjecture 1 is equivalent to the 3x+1 conjecture.
%C A257480 Theorem 4: The sequence {log(S(n))/log(n)}_{n>1} is bounded with least upper bound equal to log(3)/log(2).
%C A257480 [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.]
%D A257480 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.
%H A257480 I. Korec and Štefan Znám, <a href="http://www.jstor.org/stable/2323419">A Note on the 3x+1 Problem</a>, Amer. Math. Monthly 94, 1987, pp. 771-772.
%H A257480 J. C. Lagarias, <a href="http://www.jstor.org/stable/2322189">The 3x + 1 Problem and Its Generalizations</a>, Amer. Math. Monthly 92, 1985, pp. 3-23.
%H A257480 J. C. Lagarias, <a href="http://arxiv.org/abs/math.NT/0309224">The 3x+1 Problem: An Annotated Bibliography (1963-2000)</a>, arXiv:math/0309224 [math.NT], 2003-2011.
%H A257480 J. C. Lagarias, <a href="http://arxiv.org/abs/math/0608208">The 3x+1 Problem: an annotated bibliography, II (2000-2009)</a>, arXiv:math/0608208 [math.NT], 2006-2012.
%t A257480 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}]
%o A257480 (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
%Y A257480 Cf. A006370, A070165, A075677, A256598.
%Y A257480 Cf. A241957, A254067, A254311, A257499, A257791 (all used in the proof of Thm 4).
%Y A257480 Cf. A253676 (iteration of S terminating at the first occurrence of 1, assuming the 3x+1 conjecture).
%Y A257480 Cf. A253720, A254068, A254070, A254131, A254312, A255138, A255168, A258415.
%K A257480 nonn
%O A257480 1,3
%A A257480 _L. Edson Jeffery_, Apr 26 2015