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A008833 Largest square dividing n.

%I A008833 #85 Jul 08 2025 01:21:18
%S A008833 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,
%T A008833 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,
%U A008833 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
%N A008833 Largest square dividing n.
%C A008833 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
%H A008833 Daniel Forgues, <a href="/A008833/b008833.txt">Table of n, a(n) for n = 1..100000</a>
%H A008833 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>
%H A008833 R. J. Mathar, <a href="http://arxiv.org/abs/1106.4038">Survey of Dirichlet series of multiplicative arithmetic functions</a> arXiv:1106.4038 [math.NT], 2011-2012, Remark 16.
%H A008833 Andrew Reiter, <a href="http://www.cw-complex.com/modNspirals/">On (mod n) spirals</a>, 2014, see also posting to Number Theory Mailing List, Mar 23 2014.
%H A008833 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePart.html">Square part</a>
%F A008833 a(n) = A000188(n)^2 = n/A007913(n). Cf. A019554.
%F A008833 Multiplicative with a(p^e) = p^(2[e/2]). - _David W. Wilson_, Aug 01 2001
%F A008833 Dirichlet g.f.: zeta(s)*zeta(2s-2)/zeta(2s). - _R. J. Mathar_, Oct 31 2011
%F A008833 a(n) = A005563(n-1) / A068310(n) for n > 1. - _Reinhard Zumkeller_, Nov 26 2011
%F A008833 Sum_{k=1..n} a(k) ~ Zeta(3/2) * n^(3/2) / (3*Zeta(3)). - _Vaclav Kotesovec_, Feb 01 2019
%F A008833 a(A059897(n,k)) = A059897(a(n), a(k)). - _Peter Munn_, Nov 30 2019
%F A008833 From _Ridouane Oudra_, May 11 2025: (Start)
%F A008833 a(n) = Sum_{d|n} lambda(d)*d*psi(n/d), where lambda = A008836 and psi = A001615.
%F A008833 a(n) = lambda(n) * Sum_{d|n} lambda(d)*d*phi(n/d).
%F A008833 a(n) = A008836(n) * A358272(n). (End)
%p A008833 A008833 := proc(n)
%p A008833     expand(numtheory:-nthpow(n,2)) ;
%p A008833 end proc:
%p A008833 seq(A008833(n), n=1..100) ;
%t A008833 a[n_] := First[ Select[ Reverse[ Divisors[n]], IntegerQ[Sqrt[#]]&, 1]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Dec 12 2011 *)
%t A008833 f[p_, e_] := p^(2*Floor[e/2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Jul 07 2020 *)
%o A008833 (PARI) A008833(n)=n/core(n) \\ _Michael B. Porter_, Oct 17 2009
%o A008833 (Haskell)
%o A008833 a008833 n = head $ filter ((== 0) . (mod n)) $
%o A008833    reverse $ takeWhile (<= n) $ tail a000290_list
%o A008833 -- _Reinhard Zumkeller_, Nov 13 2011
%o A008833 (Python)
%o A008833 from sympy.ntheory.factor_ import core
%o A008833 def A008833(n): return n//core(n) # _Chai Wah Wu_, Dec 30 2021
%Y A008833 Cf. A000188, A005563, A007913, A019554, A059897, A068310.
%Y A008833 Cf. A008836, A358272, A001615.
%K A008833 nonn,easy,mult
%O A008833 1,4
%A A008833 _N. J. A. Sloane_