A114591 A composite analog of the Moebius function: Sum_{n>=1} a(n)/n^s = Product_{c=composites} (1 - 1/c^s) = zeta(s) *Product_{k>=2} (1 - 1/k^s).

1, 0, 0, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, -1, 0, 0, -1, -1, -1, -1, 0, -1, 0, 0, -1, -1, -1, 0, 0, -1, -1, 0, 0, -1, 0, -1, -1, -1, 0, 1, -1, -1, -1, -1, 0, 0, -1, 0, -1, -1, 0, 1, 0, -1, -1, 0, -1, -1, 0, -1, -1, -1, 0, 2, 0, -1, -1, -1, -1, -1, 0, 1

Offset: 1

Author

Leroy Quet, Dec 11 2005

Keywords

• sign

Comments

For n >= 2, Sum_{k|n} A050370(n/k) * a(k) = 0.

Sum_{n>=1} a(n)/n^2 = Pi^2/12.

a(n) = Sum_{k|n} A114592(k).

Examples

			24 can be factored into distinct composites as 24 and as 4*6.
So a(24) = (-1)^1 + (-1)^2 = 0, where the 1 exponent is due to the 1 factor of the 24 = 24 factorization and the 2 exponent is due to the 2 factors of the 24 = 4*6 factorization.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A045778, A050370, A114592.

Programs

Mathematica

a[n_] := Total[((-1)^Length[#]& ) /@ Select[Subsets[Select[Rest[Divisors[n]], !PrimeQ[#]& ]], Times @@ # == n & ]]; Table[a[n], {n, 1, 80}]

PARI

A114592aux(n, k) = if(1==n, 1, sumdiv(n, d, if(d > 1 && d <= k && d < n, (-1)*A114592aux(n/d, d-1))) - (n<=k)); \\ After code in A045778.
A114592(n) = A114592aux(n,n);
A114591(n) = sumdiv(n,d,A114592(d)); \\ Antti Karttunen, Jul 23 2017

Formula

a(1) = 1; for n>= 2, a(n) = sum, over ways to factor n into any number of distinct composites, of (-1)^(number of composites in a factorization). (See example.)

Extensions

More terms from Jean-François Alcover, Sep 26 2013