A033273 Number of nonprime divisors of n.

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 4, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 3, 4, 1, 5, 1, 5, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 9, 1, 2, 4, 6, 2, 5, 1, 4, 2, 5, 1, 10, 1, 2, 4, 4, 2, 5, 1, 8, 4, 2, 1, 9, 2, 2, 2, 6, 1, 9, 2, 4, 2, 2, 2, 10, 1, 4, 4, 7, 1, 5, 1, 6

Offset

1,4

Links

Ray Chandler, Table of n, a(n) for n=1..100000

Formula

a(n) = A000005(n) - A001221(n).

a((n)) = n and a(m) <> n for m < A055079(n). - Reinhard Zumkeller, Dec 16 2013

G.f.: Sum_{k>=1} (x^k - x^prime(k))/((1 - x^k)*(1 - x^prime(k))). - Ilya Gutkovskiy, Jan 17 2017

Dirichlet g.f.: zeta(s)*(zeta(s)-primezeta(s)). - Benedict W. J. Irwin, Jul 11 2018

Sum_{k=1..n} a(k) ~ n*log(n) - n*log(log(n)) + (2*gamma - 1 - B)*n, where gamma is Euler's constant (A001620) and B is Mertens's constant (A077761). - Amiram Eldar, Nov 27 2022

Mathematica

Table[Length[Select[Divisors[n], ! PrimeQ[#] &]], {n, 104}] (* Jayanta Basu, May 23 2013 *)
Table[DivisorSigma[0, n] - PrimeNu[n], {n, 100}] (* Vincenzo Librandi, May 17 2017 *)
Table[Count[Divisors[n], _?CompositeQ], {n, 110}]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 11 2019 *)

Prog

(Haskell)
a033273 = length . filter ((== 0) . a010051) . a027750_row
-- Reinhard Zumkeller, Dec 16 2013
(PARI) a(n) = numdiv(n) - omega(n); \\ Michel Marcus, Apr 28 2016
(Magma) [NumberOfDivisors(n) - #PrimeDivisors(n): n in [1..150]]; // Vincenzo Librandi, May 17 2017

Crossrefs

Cf. A000005, A001221, A010051, A027750.

Cf. A055079, A059992, A180040.

Cf. A001620, A077761.

Sequence in context: A351417 A226378 A347460 * A319685 A343652 A034836

Adjacent sequences: A033270 A033271 A033272 * A033274 A033275 A033276

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Reinhard Zumkeller, Sep 02 2003

Corrected error in offset. - Jaroslav Krizek, May 04 2009

Extended by Ray Chandler, Aug 07 2010

Status

approved