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%I #88 Sep 02 2023 07:57:11
%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. [The relation with LCM is wrong if b is not squarefree. One must, e.g., replace b with A007947(b). - _M. F. Hasler_, Mar 03 2018]
%C Instead of the terms "inner square root" and "outer square root", we may use the terms "lower square root" and "upper square root", respectively. Upper k-th roots have been studied by Broughan (2002, 2003, 2006). - _Petros Hadjicostas_, Sep 15 2019
%C The number of times each number k appears in this sequence is A034444(k). The first time k appears is at position A102631(k). - _N. J. A. Sloane_, Jul 28 2021
%H T. D. Noe, <a href="/A019554/b019554.txt">Table of n, a(n) for n = 1..10000</a>
%H Henry 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="https://doi.org/10.4064/aa101-2-2">Restricted divisor sums</a>, Acta Arithmetica, 101(2) (2002), 105-114. See also <a href="http://www.math.waikato.ac.nz/~kab/papers/div4.pdf">here</a> for another copy.
%H Kevin A. Broughan, <a href="http://ijpam.eu/contents/2003-5-3/2/2.pdf">Relationship between the integer conductor and k-th root functions</a>, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
%H Kevin A. Broughan, <a href="http://nzjm.math.auckland.ac.nz/images/d/d6/Relaxations_of_the_ABC_Conjecture_using_integer_k%27th_roots.pdf">Relaxations of the ABC Conjecture using integer k'th roots</a>, New Zealand J. Math. 35(2) (2006), 121-136.
%H Florentin 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^ceiling(e/2).
%F Dirichlet series:
%F Sum_{n>=1} a(n)/n^s = zeta(2*s-1)*zeta(s-1)/zeta(2*s-2), (Re(s) > 2);
%F Sum_{n>=1} (1/a(n))/n^s = zeta(2*s+1)*zeta(s+1)/zeta(2*s+2), (Re(s) > 0).
%F a(n) = n/A000188(n).
%F a(n) = denominator of n/n^(3/2). - _Arkadiusz Wesolowski_, Dec 04 2011
%F a(n) = Product_{k=1..A001221(n)} A027748(n,k)^ceiling(a124010(n,k)/2). - _Reinhard Zumkeller_, Apr 13 2013
%F Sum_{k=1..n} a(k) ~ 3*zeta(3)*n^2 / Pi^2. - _Vaclav Kotesovec_, Sep 18 2020
%F Sum_{k=1..n} 1/a(k) ~ 3*log(n)^2/(2*Pi^2) + (9*gamma/Pi^2 - 36*zeta'(2)/Pi^4)*log(n) + 6*gamma^2/Pi^2 - 108*gamma*zeta'(2)/Pi^4 + 432*zeta'(2)^2/Pi^6 - 36*zeta''(2)/Pi^4 - 15*sg1/Pi^2, where gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - _Vaclav Kotesovec_, Jul 27 2021
%F a(n) = sqrt(n*A007913(n)). - _Jianing Song_, May 08 2022
%F a(n) = sqrt(A053143(n)). - _Amiram Eldar_, Sep 02 2023
%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}]] (* _Harvey P. Dale_, Oct 17 2011 *)
%t f[p_, e_] := p^Ceiling[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2020 *)
%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
%o (Python 3.8+)
%o from math import prod
%o from sympy import factorint
%o def A019554(n): return n//prod(p**(q//2) for p, q in factorint(n).items()) # _Chai Wah Wu_, Aug 18 2021
%Y Cf. A000188 (inner square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).
%Y Cf. also A007913, A007947, A008833, A015049, A034444, A053143, A102631, A346602.
%K nonn,easy,mult,nice
%O 1,2
%A R. Muller
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