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A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n. 166
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

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Last modified August 13 17:37 EDT 2022. Contains 356107 sequences. (Running on oeis4.)