A309307 - OEIS

%I #13 Feb 04 2022 22:55:42

%S 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 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 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 Number of unitary divisors of n (excluding 1).

%C Also the number of squarefree divisors > 1.

%H Amiram Eldar, <a href="/A309307/b309307.txt">Table of n, a(n) for n = 1..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/UnitaryDivisor.html">Unitary Divisor</a>

%F G.f.: Sum_{k>=2} mu(k)^2*x^k/(1 - x^k).

%F Dirichlet g.f.: zeta(s)*(zeta(s)/zeta(2*s) - 1).

%F a(n) = 2^omega(n) - 1.

%F a(n) = A000225(A001221(n)) = A034444(n) - 1.

%F 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 a(n) = -1 + Sum_{d|n} mu(d)^2. - _Wesley Ivan Hurt_, Feb 04 2022

%t nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k/(1 - x^k), {k, 2, nmax}], {x, 0, nmax}], x] // Rest

%t Table[2^PrimeNu[n] - 1, {n, 1, 100}]

%Y Cf. A000225, A001221, A001222, A032741, A034444, A070824, A077610, A087893.

%K nonn

%O 1,6

%A _Ilya Gutkovskiy_, Jul 21 2019