A007949 Greatest k such that 3^k divides n. Or, 3-adic valuation of n.
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
Offset: 1
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
Crossrefs
Programs
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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 -
Magma
[Valuation(n, 3): n in [1..110]]; // Bruno Berselli, Aug 05 2013
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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:
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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 *) -
PARI
a(n)=valuation(n,3)
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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 -
Sage
[valuation(n, 3) for n in (1..106)] # Peter Luschny, Nov 16 2012
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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
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) = 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
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