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%I #6 Sep 03 2023 10:46:31
%S 1,1,1,3,1,1,1,3,4,1,1,3,1,1,1,5,1,4,1,3,1,1,1,3,6,1,4,3,1,1,1,5,1,1,
%T 1,12,1,1,1,3,1,1,1,3,4,1,1,5,8,6,1,3,1,4,1,3,1,1,1,3,1,1,4,9,1,1,1,3,
%U 1,1,1,12,1,1,6,3,1,1,1,5,10,1,1,3,1,1
%N The sum of the unitary divisors of the square root of the largest square dividing n.
%C The number of these divisors is A323308(n).
%C The sum of the unitary divisors of the largest square dividing n is A365403(n).
%H Amiram Eldar, <a href="/A365404/b365404.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A034448(A000188(n)).
%F a(n) >= 1 with equality if and only if n is squarefree (A005117).
%F Multiplicative with a(p) = 1 and a(p^e) = p^floor(e/2) + 1 for e >= 2.
%F Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-1).
%F Sum_{k=1..n} a(k) ~ (n/(2*zeta(3))) * (log(n) + 3*gamma - 1 - 4*zeta'(3)/zeta(3)), where gamma is Euler's constant (A001620).
%t f[p_, e_] := If[e == 1, 1, p^Floor[e/2] + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + f[i,1]^(f[i,2]\2)));}
%Y Cf. A005117, A000188, A034448, A323308, A365403.
%Y Cf. A001620, A002117, A244115.
%K nonn,easy,mult
%O 1,4
%A _Amiram Eldar_, Sep 03 2023
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