A038500 Highest power of 3 dividing n.

1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 27, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 9, 1, 1, 3, 1, 1, 3, 1, 1, 81

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• mult

Comments

To construct the sequence: start with 1 and concatenate twice: 1,1,1 then tripling the last term gives: 1,1,3. Concatenating those 3 terms twice gives: 1,1,3,1,1,3,1,1,3, triple the last term -> 1,1,3,1,1,3,1,1,9. Concatenating those 9 terms twice gives: 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9, triple the last term -> 1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,9,1,1,3,1,1,3,1,1,27 etc. - Benoit Cloitre, Dec 17 2002

Also 3-adic value of 1/n, n >= 1. See the Mahler reference, definition on p. 7. This is a non-archimedean valuation. See Mahler, p. 10. Sometimes also called 3-adic absolute value. - Wolfdieter Lang, Jun 28 2014

References

• Kurt Mahler, p-adic numbers and their functions, second ed., Cambridge University Press, 1981.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Tyler Ball, Tom Edgar, and Daniel Juda, Dominance Orders, Generalized Binomial Coefficients, and Kummer's Theorem, Mathematics Magazine, Vol. 87, No. 2, April 2014, pp. 135-143.
• Zoran Sunic, Tree morphisms, transducers and integer sequences, arXiv:math/0612080 [math.CO], 2006.

Crossrefs

Cf. A001620, A007949, A038502, A060904, A268354, A268357.

Programs

Haskell

a038500 = f 1 where
   f y x = if m == 0 then f (y * 3) x' else y  where (x', m) = divMod x 3
-- Reinhard Zumkeller, Jul 06 2014

Magma

[3^Valuation(n,3): n in [1..100]]; // Vincenzo Librandi, Dec 29 2015

Maple

A038500 := n -> 3^padic[ordp](n,3): # Peter Luschny, Nov 26 2010

Mathematica

Flatten[{1,1,#}&/@(3^IntegerExponent[#,3]&/@(3*Range[40]))] (* or *) hp3[n_]:=If[Divisible[n,3],3^IntegerExponent[n,3],1]; Array[hp3,90] (* Harvey P. Dale, Mar 24 2012 *)
Table[3^IntegerExponent[n, 3], {n, 100}] (* Vincenzo Librandi, Dec 29 2015 *)

PARI

{a(n) = if( n<1, 0, 3^valuation(n, 3))};

Formula

Multiplicative with a(p^e) = p^e if p = 3, 1 otherwise. - Mitch Harris, Apr 19 2005

a(n) = n / A038502(n). Dirichlet g.f. zeta(s)*(3^s-1)/(3^s-3). - R. J. Mathar, Jul 12 2012

From Peter Bala, Feb 21 2019: (Start)

a(n) = gcd(n,3^n).

O.g.f.: x/(1 - x) + 2*Sum_{n >= 1} 3^(n-1)*x^(3^n)/ (1 - x^(3^n)). (End)

Sum_{k=1..n} a(k) ~ (2/(3*log(3)))*n*log(n) + (2/3 + 2*(gamma-1)/(3*log(3)))*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 15 2022