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%I #33 Dec 18 2023 01:45:28
%S 1,1,1,2,1,1,1,1,3,1,1,2,1,1,1,4,1,3,1,2,1,1,1,1,5,1,1,2,1,1,1,2,1,1,
%T 1,6,1,1,1,1,1,1,1,2,3,1,1,4,7,5,1,2,1,1,1,1,1,1,1,2,1,1,3,8,1,1,1,2,
%U 1,1,1,3,1,1,5,2,1,1,1,4,9,1,1,2,1,1,1,1,1,3,1,2,1,1,1,2,1,7,3,10,1,1,1,1
%N Square root of largest unitary square divisor of n.
%C Unitary analog of A000188. These numbers are neither unitary nor necessarily square divisors.
%C Multiplicative because quotient of two multiplicative sequences. - _Christian G. Bower_, May 16 2005
%H Antti Karttunen, <a href="/A056622/b056622.txt">Table of n, a(n) for n = 1..16384</a>
%F a(n) = A000188(n)/A055229(n).
%F Multiplicative with a(p^e) = p^(e/2) if e even, a(p) = 1, and a(p^e) = p^((e-3)/2) for odd e > 1. - _Amiram Eldar_, Sep 14 2020
%F Dirichlet g.f.: zeta(2*s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s-1) + 1/p^(3*s)). - _Amiram Eldar_, Dec 18 2023
%e For n = 125: A000188(125) = 5, A055229(125) = 5, so a(125) = 1.
%e For n = 360: A000188(360) = 6, A055229(360) = 2, so a(360) = 3.
%t Table[Sqrt@ SelectFirst[Reverse@ Divisors@ n, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 104}] (* _Michael De Vlieger_, Dec 06 2018 *)
%t f[p_, e_] := If[EvenQ[e], p^(e/2), If[e == 1, 1, p^((e - 3)/2)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 14 2020 *)
%o (PARI)
%o A000188(n) = core(n, 1)[2]; \\ _Michel Marcus_, Feb 27 2013
%o A055229(n) = { my(c=core(n)); gcd(c, n/c); }; \\ _Charles R Greathouse IV_, Nov 20 2012
%o A056622(n) = (A000188(n)/A055229(n)); \\ _Antti Karttunen_, Nov 19 2017
%Y Cf. A000188, A055229, A034444, A056623.
%K nonn,easy,mult
%O 1,4
%A _Labos Elemer_, Aug 08 2000
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