A055212 Number of composite divisors of n.

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 3, 0, 3, 0, 3, 1, 1, 0, 5, 1, 1, 2, 3, 0, 4, 0, 4, 1, 1, 1, 6, 0, 1, 1, 5, 0, 4, 0, 3, 3, 1, 0, 7, 1, 3, 1, 3, 0, 5, 1, 5, 1, 1, 0, 8, 0, 1, 3, 5, 1, 4, 0, 3, 1, 4, 0, 9, 0, 1, 3, 3, 1, 4, 0, 7, 3, 1, 0, 8, 1, 1, 1, 5, 0, 8, 1, 3, 1, 1, 1, 9, 0, 3, 3, 6, 0, 4, 0, 5, 4

Offset: 1

Author

Leroy Quet, Jun 23 2000

Keywords

• easy
• nonn

Comments

Trivially, there is only one run of three consecutive 0's. However, there are infinitely many runs of three consecutive 1's and they are at positions A056809(n), A086005(n), and A115393(n) for n >= 1. - Timothy L. Tiffin, Jun 21 2021

Examples

			a[20] = 3 because the composite divisors of 20 are 4, 10, 20.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A083399.

Cf. A000005, A001221, A008578 (indices of 0's), A033273, A056809, A086005, A115393, A137944, A137945, A344713.

Cf. A001620, A077761.

Programs

Haskell

a055212 = subtract 1 . a033273  -- Reinhard Zumkeller, Sep 15 2015

Mathematica

Table[ Count[ PrimeQ[ Divisors[n] ], False] - 1, {n, 1, 105} ]
Table[Count[Divisors[n],?CompositeQ],{n,120}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Jul 09 2018 *)
a[n_] := DivisorSigma[0, n] - PrimeNu[n] - 1; Array[a, 100] (* Amiram Eldar, Jun 18 2022 *)

PARI

a(n) = numdiv(n) - omega(n) - 1; \\ Michel Marcus, Oct 17 2015

Formula

a(n) = A033273(n) - 1.

a(n) = tau(n)-omega(n)-1, where tau=A000005 and omega=A001221. - Reinhard Zumkeller, Jun 13 2003

G.f.: -x/(1 - x) + Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Mar 21 2017

Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 2 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Dec 07 2023