A134451 Ternary digital root of n.

0, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2

Offset: 0

Author

Reinhard Zumkeller, Oct 27 2007

Keywords

• nonn
• base
• easy
• mult

Comments

Continued fraction expansion of sqrt(3) - 1. - N. J. A. Sloane, Dec 17 2007. Cf. A040001, A048878/A002530.

Minimum number of terms required to express n as a sum of odd numbers.

Shadow transform of even numbers A005843. - Michel Marcus, Jun 06 2013

From Jianing Song, Nov 01 2022: (Start)

For n > 0, a(n) is the minimal gap of distinct numbers coprime to n. Proof: denote the minimal gap by b(n). For odd n we have A058026(n) > 0, hence b(n) = 1. For even n, since 1 and -1 are both coprime to n we have b(n) <= 2, and that b(n) >= 2 is obvious.

The maximal gap is given by A048669. (End)

Examples

			n=42: A007089(42) = '1120', A053735(42) = 1+1+2+0 = 4,
A007089(4)='11', A053735(4)=1+1=2: therefore a(42) = 2.
0.732050807568877293527446341... = 0 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...)))). - _Harry J. Smith_, May 31 2009
		

Links

• Harry J. Smith, Table of n, a(n) for n = 0..20000
• Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform.
• N. J. A. Sloane, Transforms.
• Eric Weisstein's World of Mathematics, Digital Root.
• Eric Weisstein's World of Mathematics, Ternary.

Crossrefs

Cf. A000010, A055034, A134452, A160390 (decimal expansion).

Apart from a(0) the same as A040001.

Related base-3 sequences: A053735, A134451, A230641, A230642, A230643, A230853, A230854, A230855, A230856, A230639, A230640, A010063 (trajectory of 1).

Programs

Haskell

a134451 = until (< 3) a053735
-- Reinhard Zumkeller, May 12 2011

Maple

A134451:=n->((n+1) mod 2)+2*signum(n)-1; seq(A134451(n), n=0..100); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

Table[Mod[n + 1, 2] + 2 Sign[n] - 1, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 06 2013 *)

PARI

a(n) = if(!n, 0, 2 - n%2) \\ Andrew Howroyd, Dec 08 2024

Formula

a(n) = n if n <= 2, otherwise a(A053735(n)).

a(A005408(n)) = 1; a(A005843(n)) = 2 for n>0;

a(n) = 0 if n=0, otherwise A000034(n-1).

a(n) = ((n+1) mod 2) + 2*sign(n) - 1. - Wesley Ivan Hurt, Dec 06 2013

Multiplicative with a(2^e) = 2, a(p^e) = 1 for odd prime p. - Andrew Howroyd, Aug 06 2018

a(0) = A055034(1) / A000010(1), a(n) = A000010(n+1) / A055034(n+1), n>1. - Torlach Rush, Oct 29 2019

Dirichlet g.f.: zeta(s)*(1+1/2^s). - Amiram Eldar, Jan 01 2023