%I #9 Jan 29 2023 12:10:16
%S 1,1,1,3,1,1,1,1,4,1,1,3,1,1,1,5,1,4,1,3,1,1,1,1,6,1,1,3,1,1,1,1,1,1,
%T 1,12,1,1,1,1,1,1,1,3,4,1,1,5,8,6,1,3,1,1,1,1,1,1,1,3,1,1,4,9,1,1,1,3,
%U 1,1,1,4,1,1,6,3,1,1,1,5,10,1,1,3,1,1,1
%N a(n) is the sum of the square roots of the unitary divisors of n that are squares.
%C The number of unitary divisors of n that are squares is A056624(n) and their sum is A358347(n).
%C The unitary analog of A069290.
%H Amiram Eldar, <a href="/A360162/b360162.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n, gcd(d, n/d)=1, d square} sqrt(d).
%F Multiplicative with a(p^e) = p^(e/2) + 1 if e is even, and 1 if e is odd.
%F Dirichlet g.f.: zeta(s)*zeta(2*s-1)/zeta(3*s-1).
%F Sum_{k=1..n} a(k) ~ (3*n/Pi^2)*(log(n) + 3*gamma - 1 - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).
%t f[p_, e_] := If[OddQ[e], 1, p^(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]%2, 1, f[i, 1]^(f[i, 2]/2) + 1)); }
%Y Cf. A001620, A056624, A069290, A306016, A358347.
%K nonn,easy,mult
%O 1,4
%A _Amiram Eldar_, Jan 29 2023
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