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A110529 Numbers n such that n in ternary representation (A007089) has a block of exactly a prime number of consecutive zeros. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

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.

FORMULA

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

EXAMPLE

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 sub-blocks with 2 or 3 consecutive zeros because sub-blocks 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.

MATHEMATICA

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

PROG

(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

CROSSREFS

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

Sequence in context: A332551 A121282 A205716 * A127887 A037337 A119310

Adjacent sequences:  A110526 A110527 A110528 * A110530 A110531 A110532

KEYWORD

base,easy,nonn

AUTHOR

Jonathan Vos Post, Sep 11 2005

STATUS

approved

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Last modified January 19 15:46 EST 2021. Contains 340270 sequences. (Running on oeis4.)