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A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n. 76
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

a(n) mod 2 = 1 - A014578(n). - Reinhard Zumkeller, Oct 04 2008

Obeys the general recurrences for p-adic valuation discussed in A214411. - Redjan Shabani, Jul 17 2012

Lexicographically earliest cube-free sequence, which also (conjecturally) appears in the construction of the lexicographically earliest cube-free {0,1}-sequence A282317, cf. Example section of A286940. - M. F. Hasler, May 21 2017

REFERENCES

K. Atanassov, On the 61st, 62nd and the 63rd Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 4 (1998), No. 4, 175-182.

F. Q. Gouvea, p-Adic Numbers, Springer-Verlag, 1993; see p. 23.

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

K. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.

F. Smarandache, Only Problems, Not Solutions!.

S. Northshield, An Analogue of Stern's Sequence for Z[sqrt(2)], Journal of Integer Sequences, 18 (2015), #15.11.6.

Index entries for sequences that are fixed points of mappings

FORMULA

a(n) = 0 if n != 0 (mod 3), else a(n) = 1 + a(n/3). - Reinhard Zumkeller, Aug 12 2001, edited by M. F. Hasler, Aug 11 2015

a(n) = A051064(n)-1. G.f.: Sum(k>=1, x^3^k/(1-x^3^k)))). - Ralf Stephan, Apr 12 2002

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) = 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

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

CROSSREFS

Partial sums give A054861. Cf. A080278, A001511, A122841, A007814, A112765.

Cf. A051064.

Cf. A253786.

Sequence in context: A212663 A015692 A016232 * A191265 A253786 A078595

Adjacent sequences:  A007946 A007947 A007948 * A007950 A007951 A007952

KEYWORD

nonn,easy,changed

AUTHOR

R. Muller

STATUS

approved

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Last modified May 22 21:40 EDT 2017. Contains 286906 sequences.