A055229 Greatest common divisor of largest square dividing n and squarefree part of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Author

Labos Elemer, Jun 21 2000

Keywords

• nice
• nonn
• mult

Comments

Record values occur at cubes of squarefree numbers: a(A062838(n)) = A005117(n) and a(m) < A005117(n) for m < A062838(n). - Reinhard Zumkeller, Apr 09 2010

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000188, A007913, A007947, A008833.

Cf. A027748, A005117, A062838, A060476, A124010, A220218.

Programs

Haskell

a055229 n = product $ zipWith (^) ps (map (flip mod 2) es) where
   (ps, es) = unzip $
              filter ((> 1) . snd) $ zip (a027748_row n) (a124010_row n)
-- Reinhard Zumkeller, Oct 27 2015

Mathematica

a[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]}& /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Feb 05 2014 *)

PARI

a(n)=my(c=core(n));gcd(c,n/c) \\ Charles R Greathouse IV, Nov 20 2012

Formula

a(n) = gcd[A008833(n), A007913(n)].

Multiplicative with a(p^e)=1 for even e, a(p)=1, a(p^e)=p for odd e>1. - Vladeta Jovovic, Apr 30 2002

A220218(a(n)) = 1; A060476(a(n)) > 1 for n > 1. - Reinhard Zumkeller, Nov 30 2015

a(n) = core(n)*rad(n/core(n))/rad(n), where core = A007913 and rad = A007947. - Conjecture by Velin Yanev, proof by David J. Seal, Sep 19 2017

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((p^3 + p^2 + p - 1)/(p^2 * (p + 1))) = 1.2249749939341923764... . - Amiram Eldar, Oct 08 2022