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A007949
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Greatest k such that 3^k divides n. Or, 3-adic valuation of n.
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179
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0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 3, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 4, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1
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OFFSET
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1,9
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COMMENTS
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Lexicographically earliest cubefree sequence, which also (conjecturally) appears in the construction of the lexicographically earliest cubefree {0,1}-sequence A282317, cf. Example section of A286940. - M. F. Hasler, May 21 2017
The sequence is invariant under the "lower trim" operator: remove all zeros, and subtract one from each remaining term. - Franklin T. Adams-Watters, May 25 2017
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REFERENCES
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F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} x^3^k/(1 - x^3^k). (End)
Fixed point of the morphism: 0 -> 001; 1 -> 002; 2 -> 003; 3 -> 004; 4 -> 005; etc.; starting from a(1) = 0. - Philippe Deléham, Mar 29 2004
Totally additive with a(p) = 1 if p = 3, 0 otherwise.
v_{m}(n) = Sum_{r>=1} (r/m^(r+1)) Sum_{j=1..m-1} Sum_{k=0..m^(r+1)-1} exp((2*k*Pi*i*(n+(m-j)*m^r)) / m^(r+1)). This formula is for the general case; for this specific one, set m=3. - A. Neves, Oct 04 2010
a(3n) = a(n) + 1, a(pn) = a(n) for any other prime p != 3. - M. F. Hasler, Aug 11 2015
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/2. - Amiram Eldar, Jul 11 2020
a(n) = tau(n)/(tau(3*n) - tau(n)) - 1, where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
a(n) = 3*Sum_{j=1..floor(log_3(n))} frac(binomial(n,3^j)*3^(j-1)/n). - Dario T. de Castro, Jul 10 2022
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MAPLE
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A007949 := proc(n) option remember; if n mod 3 > 0 then 0 else procname(n/3)+1; fi; end;
padic[ordp](n, 3) ;
end proc:
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MATHEMATICA
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p=3; Array[ If[ Mod[ #, p ]==0, Select[ FactorInteger[ # ], Function[ q, q[ [ 1 ] ]==p ], 1 ][ [ 1, 2 ] ], 0 ]&, 81 ]
Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 0, 1}, 1 -> {0, 0, 2}, 2 -> {0, 0, 3}, 3 -> {0, 0, 4}}) ]}], {0}, 5] (* Robert G. Wilson v, Mar 03 2005 *)
IntegerExponent[Range[200], 3] (* Zak Seidov, Apr 15 2010 *)
Table[If[Mod[n, 3] > 0, 0, 1 + b[n/3]], {n, 200}] (* Zak Seidov, Apr 15 2010 *)
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PROG
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(PARI) a(n)=valuation(n, 3)
(Haskell)
a007949 n = if m > 0 then 0 else 1 + a007949 n'
where (n', m) = divMod n 3
(MATLAB)
% Input:
% n: an integer
% Output:
% m: max power of 3 such that 3^m divides n
% M: 1-by-K matrix where M(i) is the max power of 3 such that 3^M(i) divides n
function [m, M] = Omega3(n)
M = NaN*zeros(1, n);
M(1)=0; M(2)=0; M(3)=0;
for k=4:n
if M(k-3)~=0
M(k)=M(k-k/3)+1;
else
M(k)=0;
end
end
m=M(end);
end
(Sage) [valuation(n, 3) for n in (1..106)] # Peter Luschny, Nov 16 2012
(Magma) [Valuation(n, 3): n in [1..110]]; // Bruno Berselli, Aug 05 2013
(Scheme) (define (A007949 n) (let loop ((n n) (k 0)) (cond ((not (zero? (modulo n 3))) k) (else (loop (/ n 3) (+ 1 k)))))) ;; Antti Karttunen, Oct 06 2017
(Python)
def a(n):
k = 0
while n > 0 and n%3 == 0: n //= 3; k += 1
return k
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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R. Muller
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STATUS
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approved
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