A375650 a(n) is the cardinality of the sumset of the Collatz trajectory of n.

1, 3, 23, 6, 18, 24, 69, 10, 71, 22, 68, 25, 41, 69, 125, 15, 61, 73, 104, 28, 36, 68, 110, 33, 115, 48, 3060, 69, 95, 131, 2951, 21, 133, 67, 92, 76, 108, 108, 297, 37, 3007, 45, 203, 76, 105, 117, 2914, 45, 147, 119, 183, 57, 70, 3081, 3060, 82, 228, 102, 284

Offset: 1

Author

Markus Sigg, Aug 24 2024

Keywords

• nonn

Comments

"Sumset" of a set S = {s_i} means the set of sums of pairs, s_i + s_j with i <= j.

Examples

			The Collatz trajectory of 3 is {3,10,5,16,8,4,2,1}, which has the sumset {2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,24,26,32} of size 23, so a(3) = 23.
		

Links

• Markus Sigg, Table of n, a(n) for n = 1..10000

Crossrefs

A375006 is the list of those n for which a(n) < A008908(n) * (A008908(n) + 1) / 2.

Programs

PARI

a(n) = {
  my(T = List([n]), S = Set());
  while(n > 1, n = if(n % 2 == 0, n/2, 3*n+1); listput(T, n));
  for(i = 1, #T,
    for(j = i, #T,
      S = setunion(S, Set([T[i] + T[j]]));
    )
  );
  #S
};
print(vector(59, n, a(n)));

Python

def a(n):
    T, S = [n], set()
    while n > 1:
        if n & 1 == 0: n >>= 1
        else: n = 3 * n + 1
        T.append(n)
    for i in range(len(T)):
        for j in range(i, len(T)):
            S.add(T[i] + T[j])
    return len(S)
print([a(n) for n in range(1, 60)]) # Darío Clavijo, Aug 24 2024