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%I #9 Jan 29 2023 12:07:15
%S 1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,6,1,1,1,1,1,1,1,1,1,
%T 1,4,1,1,1,1,1,1,1,1,4,1,1,1,8,6,1,1,1,1,1,1,1,1,1,1,1,1,4,1,1,1,1,1,
%U 1,1,1,4,1,1,6,1,1,1,1,1,10,1,1,1,1,1,1
%N a(n) is the sum of the square roots of the unitary divisors of n that are odd squares.
%C First differs from A336649 at n = 27.
%C The unitary analog of A360163.
%H Amiram Eldar, <a href="/A360164/b360164.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n, gcd(d, n/d)=1, d odd square} sqrt(d).
%F a(n) = A360162(n) if n is not of the form (2*m - 1)*4^k where m >= 1, k >= 1 (A108269).
%F Multiplicative with a(2^e) = 1, and for p > 2, 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))*(2^(3*s)-2^(s+1))/(2^(3*s)-2).
%F Sum_{k=1..n} a(k) ~ (2*n/Pi^2)*(log(n) + 3*gamma - 1 + log(2) - 3*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620).
%t f[p_, e_] := If[OddQ[e], 1, p^(e/2) + 1]; f[2, e_] := 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, 1] == 2, 1, if(f[i, 2]%2, 1, f[i, 1]^(f[i, 2]/2) + 1))); }
%Y Cf. A001620, A056624, A069290, A108269, A306016, A336649, A358347, A360163.
%K nonn,easy,mult
%O 1,9
%A _Amiram Eldar_, Jan 29 2023
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