A362757 The number of integers in the set f^n({0}), where f is a variant of the Collatz function that replaces any element x in the argument set with both x/2 and 3*x+1.

1, 2, 3, 5, 7, 10, 15, 22, 33, 48, 72, 103, 153, 221, 326, 477, 705, 1036, 1526, 2243, 3310, 4872, 7179, 10582, 15620, 23039, 33995, 50151, 73999, 109170, 161092, 237629, 350590, 517254, 763167, 1126070, 1661607, 2451715, 3617809, 5338044, 7876246, 11621318, 17147409, 25300982, 37331656, 55082911, 81275003

Offset: 0

Author

Markus Sigg, May 02 2023

Keywords

• nonn

Comments

a(n) is the number of integers in set A(n), where A(0) = {0} and A(n+1) = {x/2 : x in A(n)} union {3x+1 : x in A(n)}.

Non-integer numbers do not have integer offsprings. Consequently, they can be dropped when calculating terms of the sequence.

Apparently the limit of a(n)/a(n-1) is approximately equal to 1.47551 (see plot of a(n-1)/a(n) ~= 0.677732). An explanation of this limit would be desirable. - Hugo Pfoertner, May 06 2023

Examples

			a(3) = 5 is the number of integers in the set {0, 1/4, 1/2, 1, 2, 5/2, 4, 13}.
		

Links

• Markus Sigg, Table of n, a(n) for n = 0..70
• Hugo Pfoertner, Plot of ratio a(n-1)/a(n), using Plot 2.

Crossrefs

Cf. A208127, A275544.

Programs

PARI

a362757(maxn) = {
  my(A = Set([0]));
  print1(1);
  for(n = 1, maxn,
    A = setunion([t >> 1 | t <- A, bitnegimply(1,t)], [3*t+1 | t <- A]);
    print1(",", #A);
  );
};