A334992 a(n) is the smallest number larger than a(n-1) whose a(i)-th binary digit is 0 for all i<n, with a(1)=1.

1, 4, 5, 8, 9, 12, 13, 64, 65, 68, 69, 72, 73, 76, 77, 128, 129, 132, 133, 136, 137, 140, 141, 192, 193, 196, 197, 200, 201, 204, 205, 1024, 1025, 1028, 1029, 1032, 1033, 1036, 1037, 1088, 1089, 1092, 1093, 1096, 1097, 1100, 1101, 1152

Offset

1,2

Comments

If n=Sum_{i=0..t} 2^(k_i) is the binary representation of n, then a(n)=Sum_{i=0..t} 2^(A335033(k_i)) is the binary representation of a(n).

The first 2^n-1 terms are the nonzero partial sums of 2^(A335033(1)), 2^(A335033(2)), ..., 2^(A335033(n)).

2^n is in the sequence if and only if n isn't in the sequence. More specifically, 2^a(n) is not in the sequence, while 2^(A335033(n))=a(2^n).

Let F(n) be the number of terms in the sequence among 0,1,2,..,n-1. Then F(2^n)=2^(n-F(n)).

F(n)/n seems to be (1/n)^(1/log(n))^(1/loglog(n))^(1/logloglog(n))^... asymptotically (all logs are base 2).

Links

Alon Heller, Table of n, a(n) for n = 1..65536

Formula

a(Sum_{i=0..t} 2^(k_i)) = Sum_{i=0..t} 2^(A335033(k_i)) (when k_i are distinct nonnegative integers)

Example

a(2)=4 because 4 is the smallest number > 1 whose 1st binary digit is 0.
a(6)=12 because 12 is the smallest number > 9 whose 1st, 4th, 5th, 8th, and 9th binary digits are all 0.

Prog

(Python)
def gen():
  """ Generates the terms of A334992, starting with 1 """
  A334992 = [0]
  A335033 = [0]
  while True:
    new_power = 2**A335033[-1]
    for i in range(len(A334992)):
      A334992.append(A334992[i] + new_power)
      yield A334992[i] + new_power
    next_compl_elem = A335033[-1] + 1
    while next_compl_elem in A334992:
      next_compl_elem += 1
    A335033.append(next_compl_elem)
def A334992_list(n):
  """ Returns the n first elements as a list """
  g = gen()
  return [next(g) for _ in range(n)]

Crossrefs

Complement of A335033.

Sequence in context: A126001 A321333 A333384 * A269984 A188085 A316097

Adjacent sequences: A334989 A334990 A334991 * A334993 A334994 A334995

Keyword

nonn,base,easy

Author

Alon Heller, May 20 2020

Status

approved