A051064 3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.

1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 1, 2, 1, 1, 2

Offset: 1

Author

N. J. A. Sloane, Gary W. Adamson

Keywords

• nonn
• easy
• nice
• mult

Comments

a(n) is the Hamming distance between n and n-1 in ternary representation. - Philippe Deléham, Mar 29 2004

3^a(n) divides 4^n-1. - Benoit Cloitre, Oct 25 2004

Generalized Ruler Function for k=3. - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

a(A007417(n)) is odd and a(A145204(n)) is even. - Reinhard Zumkeller, May 23 2013

First n terms comprise least cubefree word of length n using positive integers, where "cubefree" means that the word contains no three consecutive identical subwords; e.g., 1 contains no cube; 11 contains no cube; 111 does but 112 does not; ... 1,1,2,1,1,2,1,1,1 does, and 1,1,2,1,1,2,1,1,2 does, but 1,1,2,1,1,2,1,1,3 does not, etc. - Clark Kimberling, Sep 10 2013

The sequence is invariant under the "lower trim" operator: remove all ones, and subtract one from each remaining term. - Franklin T. Adams-Watters, May 25 2017

a(n) is the dimension in which the coordinates of the vertices n-1 and n differ in the ternary reflected Gray code. - Arie Bos, Jul 12 2023

The number of powers of 3 that divide n. - Amiram Eldar, Mar 29 2025

Examples

			3^2 | 3*6 = 18, so a(6) = 2.
		

References

• Letter from Gary W. Adamson to N. J. A. Sloane concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• A. M. Hinz, S. Klavžar, U. Milutinović, and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 243. Book's website
• Tamas Lengyel, Divisiblity Properties by Multisection, Fib. Quart. 41 (1) (2003) 72.
• Simon Plouffe, On the values of the functions zeta and gamma, arXiv preprint arXiv:1310.7195 [math.NT], 2013.
• Joseph Rosenbaum, Elementary Problem E319, American Mathematical Monthly, volume 45, number 10, December 1938, pages 694-696. (The A indices in P at equations 1' and 2' for p=3.)
• Index entries for sequences that are fixed points of mappings.

Crossrefs

Cf. A007949.

Partial sums give A004128.

Cf. A000005, A027746.

Cf. A254046.

Cf. A001511, A055457, A115362, A373216, A373217.

Programs

Haskell

a051064 = (+ 1) . length .
                  takeWhile (== 3) . dropWhile (== 2) . a027746_row
-- Reinhard Zumkeller, May 23 2013

Maple

seq(1+padic:-ordp(n,3), n=1..100); # Robert Israel, Aug 07 2014

Mathematica

Nest[ Function[ l, {Flatten[(l /. {1 -> {1, 1, 2}, 2 -> {1, 1, 3}, 3 -> {1, 1, 4}, 4 -> {1, 1, 5}})]}], {1}, 5] (* Robert G. Wilson v, Mar 03 2005 *)
Table[ IntegerExponent[3n, 3], {n, 1, 105}] (* Jean-François Alcover, Oct 10 2011 *)

PARI

a(n)=if(n<1,0,1+valuation(n,3))

Python

def A051064(n):
    c = 1
    a, b = divmod(n,3)
    while b == 0:
        a, b = divmod(a,3)
        c += 1
    return c # Chai Wah Wu, Apr 18 2022

Formula

a(n) = A007949(n) + 1 = A004128(n) - A004128(n-1).

Multiplicative with a(p^e) = e+1 if p = 3; 1 if p <> 3. - Vladeta Jovovic, Aug 24 2002

G.f.: Sum_{k>=0} x^3^k/(1-x^3^k). - Ralf Stephan, Apr 12 2002

Fixed point of the morphism: 1 -> 112; 2 -> 113; 3 -> 114; 4 -> 115; ...; starting from a(1) = 1. a(3n+1) = a(3n+2) = 1; a(3n) = 1 + a(n). - Philippe Deléham, Mar 29 2004

a(n) = (-1)*Sum_{d divides n} mu(3d)*tau(n/d). - Benoit Cloitre, Jun 21 2007

Dirichlet g.f.: zeta(s)/(1-1/3^s). - R. J. Mathar, Jun 13 2011

a(n) = (1/2)*(3 - A053735(n) + A053735(n-1)) for n >= 1. - Tom Edgar, Aug 06 2014

a(n) = A007949(3n). - Cyril Damamme, Aug 04 2015

a(2n) = a(n), a(2n-1) = A254046(n). - Cyril Damamme, Aug 04 2015

G.f. A(x) satisfies: A(x) = A(x^3) + x/(1 - x). - Ilya Gutkovskiy, May 03 2019

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Sep 11 2020 [corrected by Vaclav Kotesovec, Jun 25 2024, see also A004128]

a(n) = tau(n)/(tau(3*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021

G.f.: Sum_{i>=1, j>=0} x^(i*3^j). - Seiichi Manyama, Mar 23 2025

Conjecture: a(n) = A007949(A000045(4*n)), all other 3-adic quadrisections A007949(A000045(.))=0. [Lengyel?]. - R. J. Mathar, Jun 28 2025

Extensions

More terms from James Sellers, Dec 11 1999

More terms from Vladeta Jovovic, Aug 24 2002