A008833 Largest square dividing n.

1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 4, 1, 1, 1, 16, 1, 9, 1, 4, 1, 1, 1, 4, 25, 1, 9, 4, 1, 1, 1, 16, 1, 1, 1, 36, 1, 1, 1, 4, 1, 1, 1, 4, 9, 1, 1, 16, 49, 25, 1, 4, 1, 9, 1, 4, 1, 1, 1, 4, 1, 1, 9, 64, 1, 1, 1, 4, 1, 1, 1, 36, 1, 1, 25, 4, 1, 1, 1, 16, 81, 1, 1, 4, 1, 1, 1, 4, 1, 9, 1, 4, 1, 1, 1, 16, 1

Offset

1,4

Comments

The Dirichlet generating function of the arithmetic function of the largest t-th power dividing n is zeta(s)*zeta(t*s-t)/zeta(s*t), here with t=2 and in A008834 and A008835 with t=3 and t=4, respectively. - R. J. Mathar, Feb 19 2011

Links

Daniel Forgues, Table of n, a(n) for n = 1..100000

Henry Bottomley, Some Smarandache-type multiplicative sequences

R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions arXiv:1106.4038 [math.NT], 2011-2012, Remark 16.

Andrew Reiter, On (mod n) spirals, 2014, see also posting to Number Theory Mailing List, Mar 23 2014.

Eric Weisstein's World of Mathematics, Square part

Formula

a(n) = A000188(n)^2 = n/A007913(n). Cf. A019554.

Multiplicative with a(p^e) = p^(2[e/2]). - David W. Wilson, Aug 01 2001

Dirichlet g.f.: zeta(s)*zeta(2s-2)/zeta(2s). - R. J. Mathar, Oct 31 2011

a(n) = A005563(n-1) / A068310(n) for n > 1. - Reinhard Zumkeller, Nov 26 2011

Sum_{k=1..n} a(k) ~ Zeta(3/2) * n^(3/2) / (3*Zeta(3)). - Vaclav Kotesovec, Feb 01 2019

a(A059897(n,k)) = A059897(a(n), a(k)). - Peter Munn, Nov 30 2019

Maple

A008833 := proc(n)
    expand(numtheory:-nthpow(n, 2)) ;
end proc:
seq(A008833(n), n=1..100) ;

Mathematica

a[n_] := First[ Select[ Reverse[ Divisors[n]], IntegerQ[Sqrt[#]]&, 1]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 12 2011 *)
f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 07 2020 *)

Prog

(PARI) A008833(n)=n/core(n) \\ Michael B. Porter, Oct 17 2009
(Haskell)
a008833 n = head $ filter ((== 0) . (mod n)) $
   reverse $ takeWhile (<= n) $ tail a000290_list
-- Reinhard Zumkeller, Nov 13 2011
(Python)
from sympy.ntheory.factor_ import core
def A008833(n): return n//core(n) # Chai Wah Wu, Dec 30 2021

Crossrefs

Cf. A000188, A005563, A007913, A019554, A059897, A068310.

Sequence in context: A366245 A083730 A358272 * A162400 A332012 A365788

Adjacent sequences: A008830 A008831 A008832 * A008834 A008835 A008836

Keyword

nonn,easy,mult

Author

N. J. A. Sloane.

Status

approved