A349407 The Farkas map: a(n) = x/3 if x mod 3 = 0; a(n) = (3x+1)/2 if x mod 3 <> 0 and x mod 4 = 3; a(n) = (x+1)/2 if x mod 3 <> 0 and x mod 4 = 1, where x = 2*n-1.

1, 1, 3, 11, 3, 17, 7, 5, 9, 29, 7, 35, 13, 9, 15, 47, 11, 53, 19, 13, 21, 65, 15, 71, 25, 17, 27, 83, 19, 89, 31, 21, 33, 101, 23, 107, 37, 25, 39, 119, 27, 125, 43, 29, 45, 137, 31, 143, 49, 33, 51, 155, 35, 161, 55, 37, 57, 173, 39, 179, 61, 41, 63, 191, 43

Offset: 1

Author

Paolo Xausa, Nov 16 2021

Keywords

• nonn
• easy

Comments

The map takes a positive odd integer x (= 2*n-1) and produces the positive odd integer a(n).

Farkas proves that the trajectory of the iterates of the map starting from any positive odd integer always reaches 1.

If displayed as a rectangular array with six columns, the columns include A016921, A016813, A016945, A004767, A239129 (see example). - Omar E. Pol, Jan 01 2022

Examples

			From _Omar E. Pol_, Jan 01 2022: (Start)
Written as a rectangular array with six columns read by rows the sequence begins:
   1,  1,  3,  11,  3,  17;
   7,  5,  9,  29,  7,  35;
  13,  9, 15,  47, 11,  53;
  19, 13, 21,  65, 15,  71;
  25, 17, 27,  83, 19,  89;
  31, 21, 33, 101, 23, 107;
  37, 25, 39, 119, 27, 125;
  43, 29, 45, 137, 31, 143;
  49, 33, 51, 155, 35, 161;
  55, 37, 57, 173, 39, 179;
...
(End)
		

References

• H. M. Farkas, "Variants of the 3N+1 Conjecture and Multiplicative Semigroups", in Entov, Pinchover and Sageev, "Geometry, Spectral Theory, Groups, and Dynamics", Contemporary Mathematics, vol. 387, American Mathematical Society, 2005, p. 121.
• J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, p. 74.

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).
• Index entries for sequences related to 3x+1 (or Collatz) problem

Crossrefs

Cf. A006370, A014682, A016813, A016921, A016945, A004767, A239129, A350279, A350515.

Programs

Mathematica

LinearRecurrence[{0,0,0,0,0,2,0,0,0,0,0,-1},{1,1,3,11,3,17,7,5,9,29,7,35},100]
Table[Which[Mod[n,3]==0,n/3,Mod[n,4]==3,(3n+1)/2,True,(n+1)/2],{n,1,200,2}] (* Harvey P. Dale, May 15 2022 *)

PARI

a(n)=if (n%3==2, 2*n\3, if (n%2, n, 3*n-1)) \\ Charles R Greathouse IV, Nov 16 2021

Python

def a(n):
    x = 2*n - 1
    return x//3 if x%3 == 0 else ((3*x+1)//2 if x%4 == 3 else (x+1)//2)
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Nov 16 2021