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A065331 Largest 3-smooth divisor of n.

%I A065331 #54 Sep 08 2024 10:02:16
%S A065331 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,
%T A065331 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,
%U A065331 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
%N A065331 Largest 3-smooth divisor of n.
%C A065331 Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) < n^0.715. - _Charles R Greathouse IV_, May 21 2014
%H A065331 Charles R Greathouse IV, <a href="/A065331/b065331.txt">Table of n, a(n) for n = 1..10000</a>
%H A065331 M. A. Bennett, M. Filaseta, and O. Trifonov, <a href="http://www.math.ubc.ca/~bennett/BFTpaper0207.pdf">On the factorization of consecutive integers</a>, J. Reine Angew. Math. 629 (2009), pp. 171-200.
%H A065331 Jon Maiga, <a href="http://sequencedb.net/s/A065331">Computer-generated formulas for A065331</a>, Sequence Machine.
%F A065331 a(n) = n / A065330(n).
%F A065331 a(n) = A006519(n) * A038500(n).
%F A065331 a(n) = (2^A007814 (n)) * (3^A007949(n)).
%F A065331 Multiplicative with a(2^e)=2^e, a(3^e)=3^e, a(p^e)=1, p>3. - _Vladeta Jovovic_, Nov 05 2001
%F A065331 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
%F A065331 a(n) = gcd(n,6^n). - _Stanislav Sykora_, Feb 08 2016
%F A065331 a(A225546(n)) = A225546(A053165(n)). - _Peter Munn_, Jan 17 2020
%F A065331 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
%F A065331 a(n) = A003586(A322026(n)), A322026(n) = A071521(a(n)). - _Antti Karttunen_, Sep 08 2024
%p A065331 A065331 := proc(n) n/A065330(n) ; end: # _R. J. Mathar_, Jun 24 2009
%p A065331 seq(2^padic:-ordp(n,2)*3^padic:-ordp(n,3), n=1..100); # _Robert Israel_, Feb 08 2016
%t A065331 Table[GCD[n, 6^n], {n, 100}] (* _Vincenzo Librandi_, Feb 09 2016 *)
%t A065331 a[n_] := Times @@ ({2, 3}^IntegerExponent[n, {2, 3}]); Array[a, 100] (* _Amiram Eldar_, Sep 19 2020 *)
%o A065331 (PARI) a(n)=3^valuation(n,3)<<valuation(n,2) \\ _Charles R Greathouse IV_, Aug 21 2011
%o A065331 (PARI) a(n)=gcd(n,6^n) \\ Not very efficient, but simple. _Stanislav Sykora_, Feb 08 2016
%o A065331 (PARI) a(n)=gcd(6^logint(n,2),n) \\ 'optimized' version of Sykora's script; _Charles R Greathouse IV_, Jul 23 2024
%o A065331 (Haskell)
%o A065331 a065331 = f 2 1 where
%o A065331    f p y x | r == 0    = f p (y * p) x'
%o A065331            | otherwise = if p == 2 then f 3 y x else y
%o A065331            where (x', r) = divMod x p
%o A065331 -- _Reinhard Zumkeller_, Nov 19 2015
%o A065331 (Magma) [Gcd(n,6^n): n in [1..100]]; // _Vincenzo Librandi_, Feb 09 2016
%Y A065331 Cf. A003586, A065330, A006519, A038500, A007814, A007949, A007310, A047229, A071521, A322026.
%Y A065331 Related to A053165 via A225546.
%Y A065331 Cf. A126760 (ordinal transform of this sequence, from its term a(1) = 1 onward).
%K A065331 mult,nonn,easy
%O A065331 1,2
%A A065331 _Reinhard Zumkeller_, Oct 29 2001