A065331 Largest 3-smooth divisor of n.

1, 2, 3, 4, 1, 6, 1, 8, 9, 2, 1, 12, 1, 2, 3, 16, 1, 18, 1, 4, 3, 2, 1, 24, 1, 2, 27, 4, 1, 6, 1, 32, 3, 2, 1, 36, 1, 2, 3, 8, 1, 6, 1, 4, 9, 2, 1, 48, 1, 2, 3, 4, 1, 54, 1, 8, 3, 2, 1, 12, 1, 2, 9, 64, 1, 6, 1, 4, 3, 2, 1, 72, 1, 2, 3, 4, 1, 6, 1, 16, 81, 2, 1, 12, 1, 2, 3, 8, 1, 18, 1, 4, 3, 2, 1, 96

Offset: 1

Author

Reinhard Zumkeller, Oct 29 2001

Keywords

• mult
• nonn
• easy

Comments

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) < n^0.715. - Charles R Greathouse IV, May 21 2014

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• M. A. Bennett, M. Filaseta, and O. Trifonov, On the factorization of consecutive integers, J. Reine Angew. Math. 629 (2009), pp. 171-200.
• Jon Maiga, Computer-generated formulas for A065331, Sequence Machine.

Crossrefs

Cf. A003586, A065330, A006519, A038500, A007814, A007949, A007310, A047229, A071521, A322026.

Related to A053165 via A225546.

Cf. A126760 (ordinal transform of this sequence, from its term a(1) = 1 onward).

Programs

Haskell

a065331 = f 2 1 where
   f p y x | r == 0    = f p (y * p) x'
           | otherwise = if p == 2 then f 3 y x else y
           where (x', r) = divMod x p
-- Reinhard Zumkeller, Nov 19 2015

Magma

[Gcd(n,6^n): n in [1..100]]; // Vincenzo Librandi, Feb 09 2016

Maple

A065331 := proc(n) n/A065330(n) ; end: # R. J. Mathar, Jun 24 2009
seq(2^padic:-ordp(n,2)*3^padic:-ordp(n,3), n=1..100); # Robert Israel, Feb 08 2016

Mathematica

Table[GCD[n, 6^n], {n, 100}] (* Vincenzo Librandi, Feb 09 2016 *)
a[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

PARI

a(n)=3^valuation(n,3)<Charles R Greathouse IV, Aug 21 2011

PARI

a(n)=gcd(n,6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016

PARI

a(n)=gcd(6^logint(n,2),n) \\ 'optimized' version of Sykora's script; Charles R Greathouse IV, Jul 23 2024

Formula

a(n) = n / A065330(n).

a(n) = A006519(n) * A038500(n).

a(n) = (2^A007814 (n)) * (3^A007949(n)).

Multiplicative with a(2^e)=2^e, a(3^e)=3^e, a(p^e)=1, p>3. - Vladeta Jovovic, Nov 05 2001

Dirichlet g.f.: zeta(s)*(1-2^(-s))*(1-3^(-s))/ ( (1-2^(1-s))*(1-3^(1-s)) ). - R. J. Mathar, Jul 04 2011

a(n) = gcd(n,6^n). - Stanislav Sykora, Feb 08 2016

a(A225546(n)) = A225546(A053165(n)). - Peter Munn, Jan 17 2020

Sum_{k=1..n} a(k) ~ n*(log(n)^2 + (2*gamma + 3*log(2) + 2*log(3) - 2)*log(n) + (2 + log(2)^2/6 + 3*log(2)*(log(3) - 1) - 2*log(3) + log(3)^2/6 + gamma*(3*log(2) + 2*log(3) - 2) - 2*sg1)) / (6*log(2)*log(3)), where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Sep 19 2020

a(n) = A003586(A322026(n)), A322026(n) = A071521(a(n)). - Antti Karttunen, Sep 08 2024