A030230 - OEIS

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

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OFFSET

1,1

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.

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

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 *)

PROG

(Haskell)

a030230 n = a030230_list !! (n-1)

a030230_list = filter (odd . a001221) [1..]

-- Reinhard Zumkeller, Aug 14 2011

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

CROSSREFS

Cf. A030231, A123066.

Cf. A008472, A007947, A014963.

Cf. A076479.