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A360157 a(n) is the number of unitary divisors of n that are odd squares.

%I A360157 #16 Mar 21 2023 15:23:16
%S A360157 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 A360157 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 A360157 1,1,1,2,1,1,2,1,1,1,1,1,2,1,1,1,1,1,1
%N A360157 a(n) is the number of unitary divisors of n that are odd squares.
%C A360157 First differs from A298735 at n = 27.
%C A360157 The unitary analog of A298735.
%C A360157 The least term that is larger than 2 is a(225) = 4.
%H A360157 Amiram Eldar, <a href="/A360157/b360157.txt">Table of n, a(n) for n = 1..10000</a>
%F A360157 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 A360157 Dirichlet g.f.: (zeta(s)*zeta(2*s)/zeta(3*s)) * (4^s + 2^s)/(4^s + 2^s + 1).
%F A360157 Sum_{k=1..n} a(k) ~ c * n, where c = Pi^2/(7*zeta(3)) = 1.172942380817... .
%F A360157 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 A360157 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 A360157 (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 A360157 Cf. A002117, A013661, A016754, A034444, A056624, A298735.
%K A360157 nonn,easy,mult
%O A360157 1,9
%A A360157 _Amiram Eldar_, Jan 29 2023