%I #16 Mar 21 2023 15:23:16
%S 1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,
%T 1,2,1,1,1,1,1,1,1,1,2,1,1,1,2,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,
%U 1,1,1,2,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1
%N a(n) is the number of unitary divisors of n that are odd squares.
%C First differs from A298735 at n = 27.
%C The unitary analog of A298735.
%C The least term that is larger than 2 is a(225) = 4.
%H Amiram Eldar, <a href="/A360157/b360157.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = 1 if e is odd and 2 if e is even.
%F Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s)/(4^s + 2^s + 1).
%F Sum_{k=1..n} a(k) ~ c * n, where c = Pi^2/(7*zeta(3)) = 1.172942380817... .
%F More precise asymptotics: Sum_{k=1..n} a(k) ~ Pi^2 * n / (7*zeta(3)) + (4 + sqrt(2)) * zeta(1/2) * sqrt(n) / (7*zeta(3/2)). - _Vaclav Kotesovec_, Jan 29 2023
%t f[p_, e_] := If[OddQ[e], 1, 2]; 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, 2]%2, 1, if(f[i, 1] == 2, 1, 2)));}
%Y Cf. A002117, A013661, A016754, A034444, A056624, A298735.
%K nonn,easy,mult
%O 1,9
%A _Amiram Eldar_, Jan 29 2023
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