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A051064
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3^a(n) exactly divides 3n. Or, 3-adic valuation of 3n.
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33
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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
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OFFSET
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1,3
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COMMENTS
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a(n) is the Hamming distance between n and n-1 in ternary representation. - Philippe Deléham, Mar 29 2004
Also : 3^a(n) divides exactly 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
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REFERENCES
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Letter from Gary W. Adamson to N. J. A. Sloane concerning Prouhet-Thue-Morse sequence, Nov. 11, 1999.
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LINKS
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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
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
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FORMULA
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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) = 3/4. - Amiram Eldar, Sep 11 2020
a(n) = tau(n)/(tau(3*n) - tau(n)), where tau(n) = A000005(n). - Peter Bala, Jan 06 2021
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EXAMPLE
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3^2 | 3*6 = 18, so a(6) = 2.
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MAPLE
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seq(1+padic:-ordp(n, 3), n=1..100); # Robert Israel, Aug 07 2014
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MATHEMATICA
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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 *)
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PROG
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(PARI) a(n)=if(n<1, 0, 1+valuation(n, 3))
(Haskell)
a051064 = (+ 1) . length .
takeWhile (== 3) . dropWhile (== 2) . a027746_row
-- Reinhard Zumkeller, May 23 2013
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CROSSREFS
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a(n) = A007949(n)+1 = A004128(n)-A004128(n-1).
Cf. A001511, A007949.
Partial sums give A004128.
Cf. A000005, A027746.
Cf. A254046.
Sequence in context: A101022 A241153 A213852 * A280509 A153096 A320010
Adjacent sequences: A051061 A051062 A051063 * A051065 A051066 A051067
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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N. J. A. Sloane, Gary W. Adamson
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EXTENSIONS
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More terms from James A. Sellers, Dec 11 1999
More terms from Vladeta Jovovic, Aug 24 2002
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STATUS
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approved
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