A038502 Remove 3's from n.

1, 2, 1, 4, 5, 2, 7, 8, 1, 10, 11, 4, 13, 14, 5, 16, 17, 2, 19, 20, 7, 22, 23, 8, 25, 26, 1, 28, 29, 10, 31, 32, 11, 34, 35, 4, 37, 38, 13, 40, 41, 14, 43, 44, 5, 46, 47, 16, 49, 50, 17, 52, 53, 2, 55, 56, 19, 58, 59, 20, 61, 62, 7, 64, 65, 22, 67, 68, 23, 70, 71, 8, 73, 74, 25, 76

Offset

1,2

Comments

As well as being multiplicative, a(n) is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. In particular, a(n) is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, Feb 21 2019

The largest divisor of n not divisible by 3. - Amiram Eldar, Sep 15 2020

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Peter Bala, A note on the sequence of numerators of a rational function .

Formula

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

a(0) = 0, a(3*n) = a(n), a(3*n+1) = 3*n+1, a(3*n+2) = 3*n+2.

Dirichlet g.f. zeta(s-1)*(3^s-3)/(3^s-1). - R. J. Mathar, Feb 11 2011

From Peter Bala, Feb 21 2019: (Start)

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

O.g.f.: F(x) - 2*F(x^3) - 2*F(x^9) - 2*F(x^27) - ..., where F(x) = x/(1 - x)^2 is the generating function for the positive integers. More generally, for m >= 1,

Sum_{n >= 0} a(n)^m*x^n = F(m,x) - (3^m - 1)( F(m,x^3) + F(m,x^9) + F(m,x^27) + ... ), where F(m,x) = A(m,x)/(1 - x)^(m+1) with A(m,x) the m_th Eulerian polynomial: A(1,x) = x, A(2,x) = x*(1 + x), A(3,x) = x*(1 + 4*x + x^2) - see A008292.

Repeatedly applying the Euler operator x*d/dx or its inverse operator to the o.g.f. for the sequence produces generating functions for the sequences n^m*a(n), m in Z. Some examples are given below. (End)

Sum_{k=1..n} a(k) ~ (3/8) * n^2. - Amiram Eldar, Oct 29 2022

a(n) = n / A038500(n). - R. J. Mathar, Mar 13 2024

Example

From Peter Bala, Feb 21 2019: (Start)
Sum_{n >= 1} n*a(n)*x^n = G(x) - (2*3)*G(x^3) - (2*9)*G(x^9) - (2*27)*G(x^27) - ..., where G(x) = x*(1 + x)/(1 - x)^3.
Sum_{n >= 1} (1/n)*a(n)*x^n = H(x) - (2/3)*H(x^3) - (2/9)*H(x^9) - (2/27)*H(x^27) - ..., where H(x) = x/(1 - x).
Sum_{n >= 1} (1/n^2)*a(n)*x^n = L(x) - (2/3^2)*L(x^3) - (2/9^2)*L(x^9) - (2/27^2)*L(x^27) - ..., where L(x) = Log(1/(1 - x)).
Also, Sum_{n >= 1} 1/a(n)*x^n = L(x) + (2/3)*L(x^3) + (2/3)*L(x^9) + (2/3)*L(x^27) + ... .
(End)

Mathematica

f[n_] := Times @@ (First@#^Last@# & /@ Select[ FactorInteger@n, First@# != 3 &]); Array[f, 76] (* Robert G. Wilson v, Jul 31 2006 *)
Table[n/3^IntegerExponent[n, 3], {n, 100}] (* Amiram Eldar, Sep 15 2020 *)

Prog

(PARI) a(n)=if(n<1, 0, n/3^valuation(n, 3)) /* Michael Somos, Nov 10 2005 */
(Haskell)
a038502 n = if m > 0 then n else a038502 n'  where (n', m) = divMod n 3
-- Reinhard Zumkeller, Jan 03 2011
(Magma) [n/3^Valuation(n, 3): n in [1..80]]; // Bruno Berselli, May 21 2013

Crossrefs

Cf. A007949, A038500, A065330.

Result of iterative removal of other factors: A000265 (2), A065883 (4), A132739 (5), A244414 (6), A242603 (7), A004151 (10).

Sequence in context: A172500 A330355 A329424 * A106610 A182398 A214736

Adjacent sequences: A038499 A038500 A038501 * A038503 A038504 A038505

Keyword

nonn,easy,mult

Author

N. J. A. Sloane

Status

approved