A065330 a(n) = max { k | gcd(n, k) = k and gcd(k, 6) = 1 }.

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, 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, 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

Offset

1,5

Comments

Bennett, Filaseta, & Trifonov show that if n > 8 then a(n^2 + n) > n^0.285. - Charles R Greathouse IV, May 21 2014

Links

Reinhard Zumkeller, 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) * A065331(n) = n.

Multiplicative with a(2^e)=1, a(3^e)=1, a(p^e)=p^e, p>3. - Vladeta Jovovic, Nov 02 2001

A106799(n) = A001222(a(n)). - Reinhard Zumkeller, May 19 2005

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

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

a(n) = A038502(A000265(n)). - Reinhard Zumkeller, Jul 06 2011

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

Sum_{k=1..n} a(k) ~ (1/4) * n^2. - Amiram Eldar, Oct 22 2022

Example

a(30) = 5.

Maple

A065330 := proc(n)
    local a, f, p, e ;
    a := 1 ;
    for f in ifactors(n)[2] do
        p := op(1, f) ;
        e := op(2, f) ;
        if p > 3 then
            a := a*p^e ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 12 2012
with(padic): a := n -> n/(2^ordp(n, 2)*3^ordp(n, 3));
seq(a(n), n=1..81); # Peter Luschny, Mar 25 2014

Mathematica

f[n_] := Times @@ (First@#^Last@# & /@ Select[FactorInteger@n, First@# != 2 && First@# != 3 &]); Array[f, 81] (* Robert G. Wilson v, Aug 18 2006 *)
f[n_]:=Denominator[6^n/n]; Array[f, 100] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2011 *)
Table[n / GCD[n, 6^n], {n, 100}] (* Vincenzo Librandi, Feb 09 2016 *)

Prog

(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
(PARI) a(n)=n\gcd(n, 6^n) \\ Not very efficient, but simple. Stanislav Sykora, Feb 08 2016
(PARI) a(n)=n>>valuation(n, 2)/3^valuation(n, 3) \\ Charles R Greathouse IV, Mar 31 2016
(Haskell)
a065330 = a038502 . a000265  -- Reinhard Zumkeller, Jul 06 2011
(Magma) [n div Gcd(n, 6^n): n in [1..100]]; // Vincenzo Librandi, Feb 09 2016

Crossrefs

Cf. A065331, A000265, A038502, A165725.

Sequence in context: A308090 A300711 A111008 * A140215 A366284 A364491

Adjacent sequences: A065327 A065328 A065329 * A065331 A065332 A065333

Keyword

mult,nonn

Author

Reinhard Zumkeller, Oct 29 2001

Status

approved