A030230 - OEIS

%I #51 Feb 12 2021 20:57:32

%S 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,

%T 53,59,60,61,64,66,67,70,71,73,78,79,81,83,84,89,90,97,101,102,103,

%U 105,107,109,110,113,114,120,121,125,126,127,128,130,131,132,137,138,139,140,149

%N Numbers that have an odd number of distinct prime divisors.

%H T. D. Noe, <a href="/A030230/b030230.txt">Table of n, a(n) for n = 1..1000</a>

%H Mats Granvik, <a href="http://pastebin.com/FJbdSsW8">Mathematica program to compute the relation to the Dirichlet inverse of the Euler totient function</a>

%H H. Helfgott and A. Ubis, <a href="https://arxiv.org/abs/1812.08707">Primos, paridad y análisis</a>, arXiv:1812.08707 [math.NT], Dec. 2018.

%F From _Benoit Cloitre_, Dec 08 2002: (Start)

%F 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).

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

%F gcd(A008472(a(n)), A007947(a(n))) > 1; see A014963. - _Labos Elemer_, Mar 26 2003

%F A076479(a(n)) = -1. - _Reinhard Zumkeller_, Jun 01 2013

%p q:= n-> is(nops(ifactors(n)[2])::odd):

%p select(q, [$1..150])[];  # _Alois P. Heinz_, Feb 12 2021

%t (* 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_ *)

%t (* Version 7.0+ *) Select[Range[2, 149], (-1)^PrimeNu[#] == -1 &]

%t Select[Range[1000],OddQ[PrimeNu[#]]&] (* _Harvey P. Dale_, Nov 27 2012 *)

%o (Haskell)

%o a030230 n = a030230_list !! (n-1)

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

%o -- _Reinhard Zumkeller_, Aug 14 2011

%o (PARI) is(n)=omega(n)%2 \\ _Charles R Greathouse IV_, Sep 14 2015

%Y Cf. A030231, A123066.

%Y Cf. A008472, A007947, A014963.

%Y Cf. A076479.

%Y Cf. A008683, A000005, A001221, A023900.

%K nonn,easy,nice

%O 1,1

%A _David W. Wilson_