A080323 -id:A080323 - cp's OEIS frontend

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

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

A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

Original entry on oeis.org

1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0, 1, 0, 0, -1, 1, -1, 0, 1, 1, 1, 0, -1, 1, 1, 0, -1, 1, -1, 0, 0, 1, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 1, 1, -1, 0, -1, 1, 0, 0, 1, 1, -1, 0, 0, 1, -1, 0, 1, 1, 1, 0, -1, 0, 1, 0, 1, 1, 1, 0, -1, 0, 0, 0, -1, 1, -1, 0, 1, 1

Offset: 1

Leroy Quet, Apr 08 2002

sign

Absolute value of a(n) = absolute value of mu(n).
Differs from A080323 at n=2, 105, 165, 195, 231, ..., 15015,..., 19635,.. (cf. A046389, A046391, ...) - R. J. Mathar, Dec 15 2008
Not multiplicative: For example a(2)*a(15) <> a(30). - R. J. Mathar, Mar 31 2012
Row products of table A225817. - Reinhard Zumkeller, Jul 30 2013

			a(6) = mu(1)*mu(2)*mu(3)*mu(6) = 1*(-1)*(-1)*1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008683, A008966, A010051, A013929, A046389, A046391, A080323, A225817.

a069158 = product . a225817_row  -- Reinhard Zumkeller, Jul 30 2013

f := function(n); t1 := &*[MoebiusMu(d) : d in Divisors(n) ]; return t1; end function;

A069158 := proc(n)
    mul(numtheory[mobius](d),d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, May 28 2016

a[n_] := Product[MoebiusMu[d], {d, Divisors[n]}]; Array[a, 106] (* Jean-François Alcover, Feb 22 2018 *)

a(n) = vecprod(apply(moebius, divisors(n))); \\ Amiram Eldar, Feb 10 2025

a(n) = 0 if mu(n) = 0 (A013929); a(n) = -1 if n = prime; a(n) = 1 if n = squarefree composite (A120944) or 1.

a(n) = A008966(n) - 2*A010051(n). - Amiram Eldar, Feb 10 2025

A080324 Union of even squarefree numbers (A039956) and squarefree numbers for which the number of prime factors is even (A030229).

Original entry on oeis.org

1, 2, 6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Reinhard Zumkeller, Feb 14 2003

nonn

k is a term of this sequence iff mu(k)^k > 0 respectively iff A080323(k) > 0.

Cf. A008683, A030229, A039956, A056912, A080323.

Definition corrected by Georg Fischer at the suggestion of Amiram Eldar, Aug 17 2023

A346773 a(n) = Sum_{d|n} möbius(d)^n.

Original entry on oeis.org

1, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 2, 0, 8, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0

Offset: 1

Seiichi Manyama, Aug 03 2021

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A008683, A023887, A034444, A068068, A080323, A100008, A264782.

Table[Sum[MoebiusMu[d]^n,{d,Divisors[n]}],{n,103}] (* Stefano Spezia, Aug 03 2021 *)

PARI
```
a(n) = sumdiv(n, d, moebius(d)^n);
```
PARI
```
a(n) = if(n%2, 0^(n-1), 2^omega(2*n));
```

N=99; x='x+O('x^N); Vec(sum(k=1, N, (moebius(k)*x)^k/(1-(moebius(k)*x)^k)))

G.f.: Sum_{k>=1} (mu(k)*x)^k/(1 - (mu(k)*x)^k).

a(2*n-1) = 0^(n-1) and a(2*n) = A034444(2*n) = A100008(n) for n > 0.

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

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A069158 a(n) = Product{d|n} mu(d), product over positive divisors, d, of n, where mu(d) = Moebius function (A008683).

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A080324 Union of even squarefree numbers (A039956) and squarefree numbers for which the number of prime factors is even (A030229).

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A346773 a(n) = Sum_{d|n} möbius(d)^n.

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