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A309307 Number of unitary divisors of n (excluding 1).

%I A309307 #14 Feb 16 2025 08:33:55
%S A309307 0,1,1,1,1,3,1,1,1,3,1,3,1,3,3,1,1,3,1,3,3,3,1,3,1,3,1,3,1,7,1,1,3,3,
%T A309307 3,3,1,3,3,3,1,7,1,3,3,3,1,3,1,3,3,3,1,3,3,3,3,3,1,7,1,3,3,1,3,7,1,3,
%U A309307 3,7,1,3,1,3,3,3,3,7,1,3,1,3,1,7,3,3,3,3,1,7,3,3,3,3,3,3,1,3,3,3
%N A309307 Number of unitary divisors of n (excluding 1).
%C A309307 Also the number of squarefree divisors > 1.
%H A309307 Amiram Eldar, <a href="/A309307/b309307.txt">Table of n, a(n) for n = 1..10000</a>
%H A309307 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>
%F A309307 G.f.: Sum_{k>=2} mu(k)^2*x^k/(1 - x^k).
%F A309307 Dirichlet g.f.: zeta(s)*(zeta(s)/zeta(2*s) - 1).
%F A309307 a(n) = 2^omega(n) - 1.
%F A309307 a(n) = A000225(A001221(n)) = A034444(n) - 1.
%F A309307 Sum_{k=1..n} a(k) ~ 6*n*(log(n) + 2*gamma - 1 - Pi^2/6 - 12*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 16 2019
%F A309307 a(n) = -1 + Sum_{d|n} mu(d)^2. - _Wesley Ivan Hurt_, Feb 04 2022
%t A309307 nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest
%t A309307 Table[2^PrimeNu[n] - 1, {n, 1, 100}]
%Y A309307 Cf. A000225, A001221, A001222, A032741, A034444, A070824, A077610, A087893.
%K A309307 nonn
%O A309307 1,6
%A A309307 _Ilya Gutkovskiy_, Jul 21 2019