A055212 - OEIS

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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

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OFFSET

1,8

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

LINKS

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

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

EXAMPLE

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

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 *)

PROG

(Haskell)

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

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

CROSSREFS

Complement of A083399.