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A008833 Largest square dividing n. 70
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 (list; graph; refs; listen; history; text; internal format)
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

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

LINKS

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

H. 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

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(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-Fran├žois 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

CROSSREFS

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

Sequence in context: A131301 A335324 A083730 * A162400 A332012 A179054

Adjacent sequences:  A008830 A008831 A008832 * A008834 A008835 A008836

KEYWORD

nonn,easy,mult,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified July 9 19:46 EDT 2020. Contains 335545 sequences. (Running on oeis4.)