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A008833
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Largest square dividing n.
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81
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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MAPLE
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A008833 := proc(n)
expand(nthpow(n, 2)) ;
end proc:
seq(A008833(n), n=1..100) ;
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000188, A005563, A007913, A019554, A059897, A068310.
Sequence in context: A131301 A335324 A083730 * A162400 A332012 A179054
Adjacent sequences: A008830 A008831 A008832 * A008834 A008835 A008836
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KEYWORD
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nonn,easy,mult
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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