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

1,2

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]

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 *)

Prog

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

Crossrefs

Intersection of A056911 with either of A000379, A028260.

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

Sequence in context: A146166 A024556 A046388 * A002557 A128907 A321644

Adjacent sequences: A056910 A056911 A056912 * A056914 A056915 A056916

Keyword

easy,nonn

Author

James A. Sellers, Jul 07 2000

Status

approved