A056913 Odd squarefree numbers for which the number of prime divisors is even.

1, 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 111, 115, 119, 123, 129, 133, 141, 143, 145, 155, 159, 161, 177, 183, 185, 187, 201, 203, 205, 209, 213, 215, 217, 219, 221, 235, 237, 247, 249, 253, 259, 265, 267, 287, 291, 295, 299, 301, 303

Offset: 1

Author

James Sellers, Jul 07 2000

Keywords

• easy
• nonn

Comments

Liouville function lambda(n) (A008836) is positive.

From Peter Munn, Jan 16 2020: (Start)

The sequence is closed under the commutative binary operation A059897(.,.). As integers are self-inverse under A059897, it forms a subgroup of the positive integers considered as a group under A059897.

This sequence is the intersection of A000379 and A056911, which are also subgroups of the positive integers under A059897.

(End)

The asymptotic density of this sequence is 2/Pi^2 (A185197). - Amiram Eldar, Oct 06 2020

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]

Crossrefs

Intersection of A056911 with either of A000379, A028260.

Cf. A056912, A059897, A008836, A026424, A185197.

Programs

Magma

[k:k in [1..303 by 2]| IsSquarefree(k) and IsEven(#PrimeDivisors(k))]; // Marius A. Burtea, Jan 21 2020

Mathematica

f[n_]:=Last/@FactorInteger[n]=={1,1}&&FactorInteger[n][[1,1]]>2; a=6;lst={1};Do[If[f[n],AppendTo[lst,n]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 23 2009 *)
Select[Range[1, 303, 2], MoebiusMu[#] == 1 &] (* Amiram Eldar, Oct 06 2020 *)

PARI

list(lim)=my(v=List([1])); forfactored(n=15,lim\1, if(n[2][1,1]>2 && vecmax(n[2][,2])==1 && #(n[2][,2])%2==0, listput(v,n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2017