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 A065331 Largest 3-smooth divisor of n. 27
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. 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 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 *) PROG (PARI) a(n)=3^valuation(n, 3)<

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