A110529 Numbers n such that n in ternary representation (A007089) has a block of exactly a prime number of consecutive zeros.

9, 18, 27, 28, 29, 36, 45, 54, 55, 56, 63, 72, 82, 83, 84, 85, 86, 87, 88, 89, 90, 99, 108, 109, 110, 117, 126, 135, 136, 137, 144, 153, 163, 164, 165, 166, 167, 168, 169, 170, 171, 180, 189, 190, 191, 198, 207, 216, 217, 218, 225, 234, 243, 246, 247, 248, 249

Offset: 1

Author

Jonathan Vos Post, Sep 11 2005

Keywords

• base
• easy
• nonn

Comments

Related to the Baum-Sweet sequence, but ternary rather than binary and prime rather than odd.

a(n) is in this sequence iff n (base 3) = A007089(n) has a block (not a subblock) of a prime number (A000040) of consecutive zeros.

Examples

			a(1) = 9 because 9 (base 3) = 100, which has a block of 2 zeros.
a(2) = 18 because 18 (base 3) = 200, which has a block of 2 zeros.
a(3) = 27 because 27 (base 3) = 1000, which has a block of 3 zeros.
81 is not in this sequence because 81 (base 3) = 10000 has a block of 4 consecutive zeros and it does not matter that this has subblocks with 2 or 3 consecutive zeros because subblocks do not count here.
243 is in this sequence because 243 (base 3) = 100000, which has a block of 5 zeros.
252 is in this sequence because 252 (base 3) = 100100 which has two blocks of 2 consecutive zeros, but we do not require there to be only one such prime-zeros block.
2187 is in this sequence because 2187 (base 3) = 10000000, which has a block of 7 zeros.
		

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.

Links

• W. Zane Billings, Table of n, a(n) for n = 1..10000
• J.-P. Allouche, Finite Automata and Arithmetic, Séminaire Lotharingien de Combinatoire, B30c (1993), 23 pp.

Crossrefs

Cf. A007089, A037011, A086747, A110471, A110472, A110474.

Programs

Mathematica

Select[Range[250], Or @@ (First[ # ] == 0 && PrimeQ[Length[ # ]] &) /@ Split[IntegerDigits[ #, 3]] &] (* Ray Chandler, Sep 12 2005 *)

Python

from re import split
from sympy import isprime
def ternary (n):
    if n == 0:
        return '0'
    nums = []
    while n:
        n, r = divmod(n, 3)
        nums.append(str(r))
    return ''.join(reversed(nums))
seq_list, n = [],1
while len(seq_list) < 10000:
    for d in split('1+|2+', ternary(n)[1:]):
        if isprime(len(d)):
            seq_list.append(n)
    n += 1
# W. Zane Billings, Jun 28 2019