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A043555
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Number of runs in base-3 representation of n.
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8
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1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 4, 3, 4, 4, 4, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 3, 4, 4, 4, 3, 4, 3, 3, 2, 2, 3, 3, 4, 3, 4, 4, 4, 3, 3, 4, 4, 3, 2, 3, 4, 4, 3, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 4, 3, 4, 4, 4, 3
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OFFSET
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0,4
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COMMENTS
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Every positive integer occurs infinitely many times. See A297770 for a guide to related sequences.
Having a(0) = 1 (rather than a(0) = 0) is debatable, on the grounds that a(0) = 1 is determined by our culture, rather than the underlying mathematics. See my August 2020 comment in A145204. - Peter Munn, Jul 12 2024
The base-2 version has a(0) = 0, corresponding to the convention that 0 has zero digits, which is the more logical (but maybe less human) convention, such that, e.g., b^n is the least number with n+1 digits in base b (<=> b^n - 1 is the largest number with n digits), valid also for 0. Here and in A043556 (base 4) the convention is that 0 has one digit, '0'.
"Runs" means substrings of consecutive equal digits, here in the base-3 representation of the numbers. See Example for details. (End)
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LINKS
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EXAMPLE
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Numbers n = 0, 1, 2, 3, 4, 5, ... are written '0', '1', '2', '10', '11', '12', ... in base 3. The first three have one single digit, so there is just 1 "run" (= subsequence of equal digits), whence a(0) = a(1) = a(2) = 1.
In '10' we have a "run" of '1's of length 1, followed by a run of '0's of length 1, so there are a(3) = 2 runs.
In '11' we have again one single run, here of 2 digits '1', whence a(4) = 1. (End)
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MAPLE
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NRUNS := proc(L::list)
local a, i;
a := 1 ;
for i from 2 to nops(L) do
if op(i, L) <> op(i-1, L) then
a := a+1 ;
end if
end do:
a ;
end proc:
convert(n, base, 3) ;
NRUNS(%) ;
end proc:
# second Maple program:
a:= n-> `if`(n<3, 1, a(iquo(n, 3))+`if`(n mod 9 in {0, 4, 8}, 0, 1)):
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MATHEMATICA
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b = 3; s[n_] := Length[Split[IntegerDigits[n, b]]];
Table[s[n], {n, 1, 200}]
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PROG
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(Python)
from itertools import groupby
from sympy.ntheory import digits
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CROSSREFS
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KEYWORD
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nonn,base,easy,changed
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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