A350515 -id:A350515 - cp's OEIS frontend

A350548 Irregular triangle T(n,k) read by rows in which row n lists the iterates of the A350515 map from n to 0.

0, 1, 0, 2, 1, 0, 3, 5, 8, 4, 1, 0, 4, 1, 0, 5, 8, 4, 1, 0, 6, 3, 5, 8, 4, 1, 0, 7, 2, 1, 0, 8, 4, 1, 0, 9, 14, 7, 2, 1, 0, 10, 3, 5, 8, 4, 1, 0, 11, 17, 26, 13, 4, 1, 0, 12, 6, 3, 5, 8, 4, 1, 0, 13, 4, 1, 0, 14, 7, 2, 1, 0, 15, 23, 35, 53, 80, 40, 13, 4, 1, 0

Offset: 0

Paolo Xausa, Jan 04 2022

			Written as an irregular triangle, the sequence begins:
  n\k   0   1   2   3   4   5   6
  -------------------------------
   0:   0
   1:   1   0
   2:   2   1   0
   3:   3   5   8   4   1   0
   4:   4   1   0
   5:   5   8   4   1   0
   6:   6   3   5   8   4   1   0
   7:   7   2   1   0
   8:   8   4   1   0
   9:   9  14   7   2   1   0
  10:  10   3   5   8   4   1   0
  11:  11  17  26  13   4   1   0
  ...

Emre Yolcu, Scott Aaronson and Marijn J. H. Heule, An Automated Approach to the Collatz Conjecture, arXiv:2105.14697 [cs.LO], 2021, pp. 21-25.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A070165, A349407, A350279, A350515.

A350515[n_]:=If[Mod[n,3]==1,(n-1)/3,If[Mod[n,6]==0||Mod[n,6]==2,n/2,(3n+1)/2]];
nrows=20;Table[NestWhileList[A350515,n,#>0&],{n,0, nrows-1}]

T(n,0) = n; T(n,k) = A350515(T(n,k-1)), where n >= 0 and k >= 1.

T(n,k) = (A350279(n+1,k+1)-1)/2, where n >= 0 and k >= 0.

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.

Original entry on oeis.org

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

Paolo Xausa, Nov 16 2021

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

			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)

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.

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

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

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 *)

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

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

A375265 a(n) = n/3 if n mod 3 = 0; otherwise a(n) = n/2 if n mod 2 = 0; otherwise a(n) = 3*n + 1.

Original entry on oeis.org

4, 1, 1, 2, 16, 2, 22, 4, 3, 5, 34, 4, 40, 7, 5, 8, 52, 6, 58, 10, 7, 11, 70, 8, 76, 13, 9, 14, 88, 10, 94, 16, 11, 17, 106, 12, 112, 19, 13, 20, 124, 14, 130, 22, 15, 23, 142, 16, 148, 25, 17, 26, 160, 18, 166, 28, 19, 29, 178, 20, 184, 31, 21, 32, 196, 22, 202, 34, 23

Offset: 1

Paolo Xausa, Aug 08 2024

Anderson (1987) reformulates the 3x+1 conjecture using this function.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Stuart Anderson, Struggling with the 3x+1 problem, The Mathematical Gazette, Vol. 71, Issue 458, December 1987, p. 273 (see A005186 for a scanned copy).
Jeffrey C. Lagarias, The 3x + 1 Problem: An Annotated Bibliography (1963-1999), arXiv:math/0309224 [math.NT], 2011, p. 6.
Index entries for sequences related to 3x+1 (or Collatz) problem.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,2,0,0,0,0,0,-1).

Cf. A375266 (trajectories).

Cf. A005186, A006370, A014682, A185452, A349407, A350515.

a := n -> ifelse(irem(n, 3) = 0, iquo(n, 3), ifelse(irem(n, 2) = 0, iquo(n, 2), 3*n + 1)): seq(a(n), n = 1..69);  # Peter Luschny, Aug 14 2024

A375265[n_] := Which[Divisible[n, 3], n/3, Divisible[n, 2], n/2, True,3*n + 1];
Array[A375265, 100]

cp's OEIS Frontend

A350548 Irregular triangle T(n,k) read by rows in which row n lists the iterates of the A350515 map from n to 0.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

A375265 a(n) = n/3 if n mod 3 = 0; otherwise a(n) = n/2 if n mod 2 = 0; otherwise a(n) = 3*n + 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica