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

1,8

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

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

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 *)

Prog

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

Crossrefs

Cf. A000188, A007913, A007947, A008833.

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

Sequence in context: A367516 A368979 A325989 * A367698 A367931 A365297

Adjacent sequences: A055226 A055227 A055228 * A055230 A055231 A055232

Keyword

nice,nonn,mult

Author

Labos Elemer, Jun 21 2000

Status

approved