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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)<<valuation(n, 2) \\ Charles R Greathouse IV, Aug 21 2011
(PARI) a(n)=gcd(n, 6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016
(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
CROSSREFS
Related to A053165 via A225546.
Sequence in context: A143771 A364255 A366283 * A066262 A195989 A174715
KEYWORD
mult,nonn,easy
AUTHOR
Reinhard Zumkeller, Oct 29 2001
STATUS
approved

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Last modified June 13 17:32 EDT 2024. Contains 373391 sequences. (Running on oeis4.)