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%I #60 Oct 22 2022 08:05:54
%S 1,1,1,1,5,1,7,1,1,5,11,1,13,7,5,1,17,1,19,5,7,11,23,1,25,13,1,7,29,5,
%T 31,1,11,17,35,1,37,19,13,5,41,7,43,11,5,23,47,1,49,25,17,13,53,1,55,
%U 7,19,29,59,5,61,31,7,1,65,11,67,17,23,35,71,1,73,37,25,19,77,13,79,5,1
%N a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.
%C Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) > n^0.285. - _Charles R Greathouse IV_, May 21 2014
%H Reinhard Zumkeller, <a href="/A065330/b065330.txt">Table of n, a(n) for n = 1..10000</a>
%H 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.
%F a(n) * A065331(n) = n.
%F Multiplicative with a(2^e)=1, a(3^e)=1, a(p^e)=p^e, p>3. - _Vladeta Jovovic_, Nov 02 2001
%F A106799(n) = A001222(a(n)). - _Reinhard Zumkeller_, May 19 2005
%F a(1)=1; then a(2n)=a(n), a(2n+1)=a((2n+1)/3) if 2n+1 is divisible by 3, a(2n+1)=2n+1 otherwise. - _Benoit Cloitre_, Jun 04 2007
%F Dirichlet g.f. zeta(s-1)*(1-2^(1-s))*(1-3^(1-s))/ ( (1-2^(-s))*(1-3^(-s)) ). - _R. J. Mathar_, Jul 04 2011
%F a(n) = A038502(A000265(n)). - _Reinhard Zumkeller_, Jul 06 2011
%F a(n) = n/GCD(n,6^n). - _Stanislav Sykora_, Feb 08 2016
%F Sum_{k=1..n} a(k) ~ (1/4) * n^2. - _Amiram Eldar_, Oct 22 2022
%e a(30) = 5.
%p A065330 := proc(n)
%p local a,f,p,e ;
%p a := 1 ;
%p for f in ifactors(n)[2] do
%p p := op(1,f) ;
%p e := op(2,f) ;
%p if p > 3 then
%p a := a*p^e ;
%p end if;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Jul 12 2012
%p with(padic): a := n -> n/(2^ordp(n, 2)*3^ordp(n, 3));
%p seq(a(n), n=1..81); # _Peter Luschny_, Mar 25 2014
%t f[n_] := Times @@ (First@#^Last@# & /@ Select[FactorInteger@n, First@# != 2 && First@# != 3 &]); Array[f, 81] (* _Robert G. Wilson v_, Aug 18 2006 *)
%t f[n_]:=Denominator[6^n/n];Array[f,100] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2011 *)
%t Table[n / GCD[n, 6^n], {n, 100}] (* _Vincenzo Librandi_, Feb 09 2016 *)
%o (PARI) a(n)=if(n<2,1,if(n%2,if(n%3,n,a(n/3)),a(n/2))) \\ _Benoit Cloitre_, Jun 04 2007
%o (PARI) a(n)=n\gcd(n,6^n) \\ Not very efficient, but simple. _Stanislav Sykora_, Feb 08 2016
%o (PARI) a(n)=n>>valuation(n,2)/3^valuation(n,3) \\ _Charles R Greathouse IV_, Mar 31 2016
%o (Haskell)
%o a065330 = a038502 . a000265 -- _Reinhard Zumkeller_, Jul 06 2011
%o (Magma) [n div Gcd(n, 6^n): n in [1..100]]; // _Vincenzo Librandi_, Feb 09 2016
%Y Cf. A065331, A000265, A038502, A165725.
%K mult,nonn
%O 1,5
%A _Reinhard Zumkeller_, Oct 29 2001
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