A030230 Numbers that have an odd number of distinct prime divisors.

2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 30, 31, 32, 37, 41, 42, 43, 47, 49, 53, 59, 60, 61, 64, 66, 67, 70, 71, 73, 78, 79, 81, 83, 84, 89, 90, 97, 101, 102, 103, 105, 107, 109, 110, 113, 114, 120, 121, 125, 126, 127, 128, 130, 131, 132, 137, 138, 139, 140, 149

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy
• nice

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Mats Granvik, Mathematica program to compute the relation to the Dirichlet inverse of the Euler totient function
• H. Helfgott and A. Ubis, Primos, paridad y análisis, arXiv:1812.08707 [math.NT], Dec. 2018.

Crossrefs

Cf. A030231, A123066.

Cf. A008472, A007947, A014963.

Cf. A076479.

Cf. A008683, A000005, A001221, A023900.

Programs

Haskell

a030230 n = a030230_list !! (n-1)
a030230_list = filter (odd . a001221) [1..]
-- Reinhard Zumkeller, Aug 14 2011

Maple

q:= n-> is(nops(ifactors(n)[2])::odd):
select(q, [$1..150])[];  # Alois P. Heinz, Feb 12 2021

Mathematica

(* Prior to version 7.0 *) littleOmega[n_] := Length[FactorInteger[n]]; Select[ Range[2, 149], (-1)^littleOmega[#] == -1 &] (* Jean-François Alcover, Nov 30 2011, after Benoit Cloitre *)
(* Version 7.0+ *) Select[Range[2, 149], (-1)^PrimeNu[#] == -1 &]
Select[Range[1000],OddQ[PrimeNu[#]]&] (* Harvey P. Dale, Nov 27 2012 *)

PARI

is(n)=omega(n)%2 \\ Charles R Greathouse IV, Sep 14 2015

Formula

From Benoit Cloitre, Dec 08 2002: (Start)

k such that Sum_{d|k} mu(d)*tau(d) = (-1)^omega(k) = -1 where mu(d) = A008683(d), tau(d) = A000005(d) and omega(d) = A001221(d).

k such that A023900(k) < 0. (End)

gcd(A008472(a(n)), A007947(a(n))) > 1; see A014963. - Labos Elemer, Mar 26 2003

A076479(a(n)) = -1. - Reinhard Zumkeller, Jun 01 2013