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%I
%S 1,2,3,2,5,6,7,4,3,10,11,6,13,14,15,4,17,6,19,10,21,22,23,12,5,26,9,
%T 14,29,30,31,8,33,34,35,6,37,38,39,20,41,42,43,22,15,46,47,12,7,10,51,
%U 26,53,18,55,28,57,58,59,30,61,62,21,8,65,66,67,34,69,70,71,12,73,74,15,38,77
%N Smallest number whose square is divisible by n.
%C A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), LCM(b,c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n.
%H T. D. Noe, <a href="/A019554/b019554.txt">Table of n, a(n) for n = 1..10000</a>
%H H. Bottomley, <a href="http://fs.gallup.unm.edu/Bottomley-Sm-Mult-Functions.htm">Some Smarandache-type multiplicative sequences</a>
%H Kevin A. Broughan, <a href="http://www.math.waikato.ac.nz/~kab/papers/div4.pdf">Restricted divisor sums</a>, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/CP2.pdf">Collected Papers, Vol. II</a>, Tempus Publ. Hse, Bucharest, 1996.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SmarandacheCeilFunction.html">Smarandache Ceil Function</a>
%F Replace any square factors in n by their square roots.
%F Multiplicative with a(p^e) = p^ceil(e/2). Dirichlet series: Sum(n=1..inf, a(n)/n^s) = zeta(2*s-1)*zeta(s-1)/zeta(2*s-2), (Re(s)>2); Sum(n=1..inf, (1/a(n))/n^s) = zeta(2*s+1)*zeta(s+1)/zeta(2*s+2), (Re(s)>0)
%F a(n) = denominator of n/n^(3/2). [Arkadiusz Wesolowski, Dec 04 2011]
%F a(n) = product(A027748(n,k)^ceiling(a124010(n,k)/2): k=1..A001221(n)). - _Reinhard Zumkeller_, Apr 13 2013
%p with(numtheory):A019554 := proc(n) local i: RETURN(op(mul(i,i=map(x->x[1]^ceil(x[2]/2),ifactors(n)[2])))); end;
%t Flatten[Table[Select[Range[n],Divisible[#^2,n]&,1],{n,100}]] (* From Harvey P. Dale, Oct 17 2011 *)
%o (PARI) a(n)=n/core(n,1)[2] \\ _Charles R Greathouse IV_, Feb 24, 2011
%o (Haskell)
%o a019554 n = product $ zipWith (^)
%o (a027748_row n) (map ((`div` 2) . (+ 1)) $ a124010_row n)
%o -- _Reinhard Zumkeller_, Apr 13 2013
%Y Cf. A019555, A008833, A015049, A000188 A007913 A007947.
%Y a(n) = n/A000188(n)
%K nonn,easy,mult,nice
%O 1,2
%A R. Muller
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