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

1,6

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

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

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}]

Crossrefs

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

Sequence in context: A243915 A367482 A367095 * A325446 A337322 A090176

Adjacent sequences: A309304 A309305 A309306 * A309308 A309309 A309310

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 21 2019

Status

approved