|
| |
|
|
A007949
|
|
Greatest k such that 3^k divides n. Or, 3-adic valuation of n.
|
|
171
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,9
|
|
|
COMMENTS
|
Obeys the general recurrences for p-adic valuation discussed in A214411. - Redjan Shabani, Jul 17 2012
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
|
|
|
REFERENCES
|
F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.
|
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n = 1..1000
K. Atanassov, On the 61-st, 62-nd and the 63-rd Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4 (1998), No. 4, 175-182.
K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
Dario T. de Castro, P-adic Order of Positive Integers via Binomial Coefficients, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 22, Paper A61, 2022.
S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.
F. Smarandache, Only Problems, Not Solutions!.
M. Vassilev-Missana and K. Atanassov, Some Representations related to n!, Notes on Number Theory and Discrete Mathematics, Vol. 4 (1998), No. 4, 148-153.
Index entries for sequences that are fixed points of mappings
|
|
|
FORMULA
|
a(n) = 0 if n != 0 (mod 3), otherwise a(n) = 1 + a(n/3). - Reinhard Zumkeller, Aug 12 2001, edited by M. F. Hasler, Aug 11 2015
From Ralf Stephan, Apr 12 2002: (Start)
a(n) = A051064(n) - 1.
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
a(n) mod 2 = 1 - A014578(n). - Reinhard Zumkeller, Oct 04 2008
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) = A051064(n), a(2n) = a(n), a(2n-1) = A253786(n). - Cyril Damamme, Aug 04 2015
a(3n) = a(n) + 1, a(pn) = a(n) for any other prime p != 3. - M. F. Hasler, Aug 11 2015
3^a(n) = A038500(n). - Antti Karttunen, Oct 09 2017
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
|
|
|
MAPLE
|
A007949 := proc(n) option remember; if n mod 3 > 0 then 0 else procname(n/3)+1; fi; end;
# alternative by R. J. Mathar, Mar 29 2017
A007949 := proc(n)
padic[ordp](n, 3) ;
end proc:
|
|
|
MATHEMATICA
|
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 *)
|
|
|
PROG
|
(PARI) a(n)=valuation(n, 3)
(Haskell)
a007949 n = if m > 0 then 0 else 1 + a007949 n'
where (n', m) = divMod n 3
-- Reinhard Zumkeller, Jun 23 2013, May 14 2011
(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
% Redjan Shabani, Jul 17 2012
(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
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Aug 06 2021
|
|
|
CROSSREFS
|
Partial sums give A054861.
Cf. A000005, A038500, A080278, A001511, A122841, A007814, A112765, A253786.
One less than A051064.
Sequence in context: A341774 A015692 A016232 * A191265 A320003 A291749
Adjacent sequences: A007946 A007947 A007948 * A007950 A007951 A007952
|
|
|
KEYWORD
|
nonn,easy
|
|
|
AUTHOR
|
R. Muller
|
|
|
STATUS
|
approved
|
| |
|
|
