A168607 -id:A168607 - cp's OEIS frontend

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

Gennady Eremin, Jan 11 2021

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.

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.

Intersection of A001651 and A340131.
Subsequences: A134752, A168607.
Cf. A026107.

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)

A337837 Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

Original entry on oeis.org

2, 4, 12, 18, 20, 28, 30, 31, 34, 35, 38, 44, 45, 49, 50, 58, 60, 75, 79, 97, 100, 103, 111, 113, 118, 120, 135, 141, 153, 154, 156, 166, 168, 171, 178, 181, 204, 219, 220, 239, 245, 247, 254, 260, 267, 269, 280, 286, 298, 307, 313

Offset: 1

Zak Seidov, Sep 25 2020

nonn

The corresponding values of Omega: 1, 1, 2, 3, 2, 2, 3, 3, 3, 3, 4, 2, 3, 3, 3, 3, 2, 6, 5, 4, 3, 4, 4, 4, 2, 4, 3, 3, 7, 4, 2, 4, 4, 4, 4, 5, 5, 5, 3, 5, 5, 6, 5, 6, 4, 5, 4, 5, 7, 6, 8.

			2 is a term since Omega(3^2 - 2) = Omega(7) = 1, and Omega(3^2 + 2) = Omega(11) = 1.

Cf. A001222, A014224, A051783, A058481, A168607.

Select[Range[200],PrimeOmega[3^#-2] == PrimeOmega[3^#+2]&]

for (k = 1, 200, if ((m = bigomega (3^k - 2)) == bigomega (3^k + 2), print (k ", " m ", ")))

a(36)-a(51) from Amiram Eldar, Sep 25 2020

A375324 Numbers of the form 3^k + 2 that admit at least one divisor of the form 3^m + 2 with 1 <= m < k.

Original entry on oeis.org

245, 2189, 19685, 531443, 1594325, 129140165, 10460353205, 31381059611, 847288609445, 7625597484989, 68630377364885, 617673396283949, 1853020188851843, 5559060566555525, 450283905890997365, 36472996377170786405, 109418989131512359211, 2954312706550833698645

Offset: 1

Marius A. Burtea, Sep 15 2024

nonn

The sequence is inspired by problem 3, Balkan Mathematical Olympiad 27 April - 2 May 2024, Varna, Bulgaria, (see link).

The sequence is infinite because numbers of the form m = 3^(4*k + 1) + 2 are divisible by 5 = 3^1 + 2.

			245 = 3^5 + 2  and 245 = 49*5 = 49 * (3^1 + 2), so 245 is a term.
2189 = 3^7 + 2  and 2189 = 11*199 = 199 * (3^2 + 2), so 2189 is a term.
129140165 = 3^17 + 2 and 129140165 = 5*25828033 = (3^1 + 2)*25828033 or 129140165 = 11*11740015 = (3^2 + 2)*11740015, so 129140165 is a term.

Balkan Mathematical Olympiad, 2024, Problems

Cf. A168607.

f:=func; [n:n in [3^a+2:a in [1..50]]|exists{d: d in Divisors(n)|d ne n and f(d-2) }];

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A340544 Numbers from A340131 that are not multiples of 3.

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A337837 Numbers k such that Omega(3^k - 2) = Omega(3^k + 2) where Omega is A001222.

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A375324 Numbers of the form 3^k + 2 that admit at least one divisor of the form 3^m + 2 with 1 <= m < k.

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