A002556 -id:A002556 - cp's OEIS frontend

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

James Sellers, Jul 07 2000

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

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]

Intersection of A056911 with either of A000379, A028260.

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

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

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 *)

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

A056912 Odd squarefree numbers for which the number of prime divisors is odd.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 105, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 165, 167, 173, 179, 181, 191, 193, 195, 197, 199, 211, 223, 227, 229, 231, 233, 239, 241, 251, 255

Offset: 1

James Sellers, Jul 07 2000

Liouville function lambda(n) (A008836) is negative.
m is a term iff mu(m)^m < 0 (A080323(a(n))<0), where mu is the Moebius function (A008683). - Reinhard Zumkeller, Feb 14 2003
The asymptotic density of this sequence is 2/Pi^2 (A185197). - Amiram Eldar, Oct 06 2020

			a(27) = 3*5*7 = 105 is the least nonprime.

H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.

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]

Cf. A056911, A056913, A008836, A026424, A028260, A185197.

Select[Range[3, 300], SquareFreeQ[#] && LiouvilleLambda[#] == -1 &] (* Jean-François Alcover, Jul 30 2013 *)
Select[Range[1, 255, 2], MoebiusMu[#] == -1 &] (* Amiram Eldar, Oct 06 2020 *)

isok(n) = (n%2) && issquarefree(n) && (omega(n)%2) \\ Michel Marcus, Jun 15 2013

is(n)=if(n%2, my(f=factor(n)[,2]);n>1 && vecmax(f)<2 && #f%2, 0) \\ Charles R Greathouse IV, Jun 15 2013

A002557 Odd squarefree numbers with an even number of prime factors that have no prime factors greater than 31.

Original entry on oeis.org

1, 15, 21, 33, 35, 39, 51, 55, 57, 65, 69, 77, 85, 87, 91, 93, 95, 115, 119, 133, 143, 145, 155, 161, 187, 203, 209, 217, 221, 247, 253, 299, 319, 323, 341, 377, 391, 403, 437, 493, 527, 551, 589, 667, 713, 899, 1155, 1365, 1785, 1995, 2145, 2415, 2805, 3003

Offset: 1

N. J. A. Sloane

Original name: A subset of A056913, definition unclear.
The definition is given on page 70 of Gupta (1943), but is hard to understand.
A variant of A056913, which has terms that also have prime factors > 31. - Arkadiusz Wesolowski, Jan 21 2016
The b-file contains the full sequence. - Robert Israel, Jan 21 2016
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. A subgroup of A056913. - Peter Munn, Jan 16 2020

H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert Israel, Table of n, a(n) for n = 1..512
H. Gupta, A formula for L(n), J. Indian Math. Soc., 7 (1943), 68-71. [Annotated scanned copy]

Cf. A002556, A046337, A059897. Subset of A056913.

a:= func< n | Factorization(n)>; [1] cat [n: n in [3..3003 by 2] | IsSquarefree(n) and (-1)^&+[p[2]: p in a(n)] eq 1 and f[#f][1] le 31 where f is a(n)]; // Arkadiusz Wesolowski, Jan 21 2016

S:= select(t -> (nops(t)::even), combinat:-powerset(select(isprime, [seq(i,i=3..31,2)]))):
sort(map(convert,S,`*`)); # Robert Israel, Jan 21 2016

npfQ[n_]:=With[{pf=FactorInteger[n][[;;,1]]},SquareFreeQ[n]&&EvenQ[PrimeOmega[n]]&&Max[pf]<32]; Select[Range[1,3003,2],npfQ] (* Harvey P. Dale, May 03 2025 *)

powerset = lambda lst: reduce(lambda result, x: result + [subset + [x] for subset in result], lst, [[]])
product = lambda lst: reduce(lambda x, y: x*y, lst, 1)
primes = [3, 5, 7, 11, 13, 17, 19, 23, 29, 31]
sequence = sorted(product(s) for s in powerset(primes) if len(s) % 2 == 0) # David Radcliffe, Jan 21 2016

Name changed and sequence extended by Arkadiusz Wesolowski, Jan 21 2016

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

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A056912 Odd squarefree numbers for which the number of prime divisors is odd.

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A002557 Odd squarefree numbers with an even number of prime factors that have no prime factors greater than 31.

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