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

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, 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, 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

Offset: 1

Author

Ilya Gutkovskiy, Jul 21 2019

Keywords

• nonn

Comments

Also the number of squarefree divisors > 1.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Unitary Divisor

Crossrefs

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

Programs

Mathematica

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

Formula

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

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

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

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

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

a(n) = -1 + Sum_{d|n} mu(d)^2. - Wesley Ivan Hurt, Feb 04 2022