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A055491 Smallest square divisible by n divided by largest square which divides n.

%I A055491 #28 Sep 16 2023 05:33:44
%S A055491 1,4,9,1,25,36,49,4,1,100,121,9,169,196,225,1,289,4,361,25,441,484,
%T A055491 529,36,1,676,9,49,841,900,961,4,1089,1156,1225,1,1369,1444,1521,100,
%U A055491 1681,1764,1849,121,25,2116,2209,9,1,4,2601,169,2809,36,3025,196,3249,3364
%N A055491 Smallest square divisible by n divided by largest square which divides n.
%H A055491 Reinhard Zumkeller, <a href="/A055491/b055491.txt">Table of n, a(n) for n = 1..10000</a>
%H A055491 Henry Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>.
%F A055491 If n is written as Product(Pj^Ej) then a(n) = Product(Pj^(2*(Ej mod 2))).
%F A055491 a(n) = A053143(n)/A008833(n) = A007913(n)^2 = (A019554(n)/A000188(n))^2 = A000290(n)/A008833(n)^2.
%F A055491 Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(6)/(3*zeta(2))) = 2*Pi^4/945 = 0.206156... . - _Amiram Eldar_, Oct 27 2022
%F A055491 Dirichlet g.f.: zeta(s-2) * zeta(2*s) / zeta(2*s-4). - _Amiram Eldar_, Sep 16 2023
%e A055491 a(12) = 36/4 = 9.
%t A055491 With[{sqs=Range[100]^2},Table[SelectFirst[sqs,Divisible[#,n]&]/ SelectFirst[ Reverse[sqs],Divisible[n,#]&],{n,60}]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Feb 18 2018 *)
%t A055491 f[p_, e_] := p^(2 * Mod[e, 2]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 07 2020 *)
%o A055491 (Haskell)
%o A055491 a055491 = (^ 2) . a007913  -- _Reinhard Zumkeller_, Jul 23 2014
%o A055491 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^(2*(f[i,2]%2)));} \\ _Amiram Eldar_, Oct 27 2022
%Y A055491 Cf. A056551, A056552.
%Y A055491 Cf. A000188, A000290, A007913, A008833, A019554, A053143.
%Y A055491 Cf. A013661, A013664.
%K A055491 easy,nonn,mult
%O A055491 1,2
%A A055491 _Henry Bottomley_, Jun 28 2000