

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
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OFFSET

1,1


COMMENTS

Related to the BaumSweet 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 subblock) 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 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 primezeros 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



