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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
OFFSET
1,1
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.
REFERENCES
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.
LINKS
J.-P. Allouche, Finite Automata and Arithmetic, Séminaire Lotharingien de Combinatoire, B30c (1993), 23 pp.
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 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
KEYWORD
base,easy,nonn
AUTHOR
Jonathan Vos Post, Sep 11 2005
STATUS
approved