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A139391 Next odd term in Collatz trajectory with starting value n.

%I A139391 #45 Apr 12 2025 16:05:38
%S A139391 1,1,5,1,1,3,11,1,7,5,17,3,5,7,23,1,13,9,29,5,1,11,35,3,19,13,41,7,11,
%T A139391 15,47,1,25,17,53,9,7,19,59,5,31,21,65,11,17,23,71,3,37,25,77,13,5,27,
%U A139391 83,7,43,29,89,15,23,31,95,1,49,33,101,17,13,35,107,9,55,37,113,19,29
%N A139391 Next odd term in Collatz trajectory with starting value n.
%H A139391 Reinhard Zumkeller, <a href="/A139391/b139391.txt">Table of n, a(n) for n = 1..10000</a>
%H A139391 Friedrich L. Bauer, <a href="https://doi.org/10.1007/s00287-008-0231-7">Der (ungerade) Collatz-Baum</a>, Informatik Spektrum 31 (Springer, April 2008).
%H A139391 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>.
%H A139391 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>.
%H A139391 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>.
%F A139391 a(n) = A006370(n) if A006370(n) is odd, otherwise a(A006370(n)).
%F A139391 a(n) = A006370(n) iff n mod 4 = 2;
%F A139391 a(A016825(n)) = A006370(A016825(n));
%F A139391 a(n) = A000265(A006370(n)).
%F A139391 a(A160967(n)) = 1. - _Reinhard Zumkeller_, May 31 2009
%F A139391 For odd n, a(n) = a(2*A350091((n-1)/2)+1). - _Ruud H.G. van Tol_, Dec 17 2021
%F A139391 Sum_{k=1..n} a(k) ~ n^2 / 3. - _Amiram Eldar_, Aug 26 2024
%F A139391 a(n) = A000265(A014682(n)). - _Alan Michael Gómez Calderón_, Apr 10 2025
%t A139391 a[n_]:=Select[NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &],OddQ]; Prepend[Table[If[EvenQ[n],a[n][[1]],a[n][[2]]],{n,2,77}],1] (* _Jayanta Basu_, May 27 2013 *)
%o A139391 (Python) # first formula
%o A139391 def A006370(n): return 3*n+1 if n%2 else n//2
%o A139391 def a(n): return x if (x := A006370(n))%2 else a(x)
%o A139391 print([a(n) for n in range(1, 78)]) # _Michael S. Branicky_, Dec 15 2021
%o A139391 (Python) # fourth formula, uses A006370 above
%o A139391 def A000265(n):
%o A139391     while n%2 == 0: n //= 2
%o A139391     return n
%o A139391 def a(n): return A000265(A006370(n))
%o A139391 print([a(n) for n in range(1, 78)]) # _Michael S. Branicky_, Dec 15 2021
%o A139391 (PARI) a(n) = my(x = if(n%2, 3*n+1, n/2)); x/2^valuation(x, 2); \\ _Michel Marcus_, Feb 27 2022
%Y A139391 Cf. A000265, A006370, A014682, A016825, A160967, A350091.
%Y A139391 Cf. A075677 (odd bisection).
%K A139391 nonn,easy
%O A139391 1,3
%A A139391 _Reinhard Zumkeller_, Apr 17 2008