A340544 Numbers from A340131 that are not multiples of 3.

5, 11, 29, 44, 50, 83, 98, 104, 116, 128, 140, 146, 245, 260, 266, 278, 290, 302, 308, 332, 344, 377, 380, 395, 401, 410, 416, 434, 449, 455, 731, 746, 752, 764, 776, 788, 794, 818, 830, 863, 866, 881, 887, 896, 902, 920, 935, 941, 980, 992, 1025, 1028, 1043

Offset

1,1

Comments

Terms are reduced, i.e., ternary codes do not have trailing zeros.

The term is a digitized Motzkin path that starts with an up step and ends with a down step. Such a path has neither leading nor final flat steps, i.e., the ternary code of the corresponding term has no finite 0's. Recall that in ternary code, 1's are up steps, and 2's are down steps.

The number of terms with a ternary code of length k is A026107(k-1). For instance, 7 (seven) reduced terms 83, 98, 104, 116, 128, 140, and 146 have a ternary length of 5, namely 10002, 10122, 10212, 11022, 11202, 12012, and 12102. Respectively A026107(4) = 7.

Links

Gennady Eremin, Table of n, a(n) for n = 1..1000

Kurt Mahler, The representation of squares to the base 3, Acta Arith. Vol. 53, Issue 1 (1989), p. 105.

Prog

(Python)
def digits(n, b):
  out = []
  while n >= b:
    out.append(n % b)
    n //= b
  return [n] + out[::-1]
def ok(n):
  if n%3 == 0: return False
  t = digits(n, 3)
  if t.count(1) != t.count(2): return False
  return all(t[:i].count(1) >= t[:i].count(2) for i in range(1, len(t)))
print([n for n in range(750) if ok(n)]) # after Michael S. Branicky (A340131)

Crossrefs

Intersection of A001651 and A340131.

Subsequences: A134752, A168607.

Cf. A026107.

Sequence in context: A088486 A174917 A214451 * A062772 A319597 A030080

Adjacent sequences: A340541 A340542 A340543 * A340545 A340546 A340547

Keyword

nonn,easy,base

Author

Gennady Eremin, Jan 11 2021

Status

approved