A349955 Numbers whose representation in any base b >= 2 is a cubefree word.

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 18, 19, 20, 22, 25, 36, 37, 38, 44, 45, 50, 51, 52, 74, 75, 76, 77, 89, 90, 100, 101, 102, 105, 109, 147, 150, 153, 154, 165, 166, 173, 178, 179, 180, 181, 204, 205, 210, 214, 217, 293, 294, 300, 301, 306, 308, 309, 329, 330

Offset: 1

Author

Vladimir Reshetnikov, Mar 20 2022

Keywords

• nonn
• base

Comments

A subsequence of A178905. A subsequence of A286262.

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Cubefree Word.

Crossrefs

Cf. A178905, A286262.

Programs

Mathematica

Prepend[Cases[Range[330], n_ /; NoneTrue[Range[2, (Sqrt[4 n - 3] - 1)/2], MatchQ[IntegerDigits[n, #], {_, d__, d__, d__, _}] &]], 0]

Python

from sympy.ntheory.digits import digits
def hascube(s):
    for l in range(1, len(s)//3 + 1):
      for i in range(len(s) - 3*l + 1):
          if s[i:i+l] == s[i+l:i+2*l] == s[i+2*l:i+3*l]: return True
    return False
def ok(n):
    if n < 7: return True
    b = 2
    d = digits(n, b)[1:]
    while len(d) >= 3:
        if hascube(d):
            return False
        b += 1
        d = digits(n, b)[1:]
    return True
print([k for k in range(331) if ok(k)]) # Michael S. Branicky, Mar 27 2022