A118206 Euler transform of the Liouville function.

1, 1, 0, -1, 0, 0, 0, -1, -1, 0, 2, 0, -2, -2, 1, 2, 2, -2, -2, 0, 2, -1, -1, -2, 2, 5, 4, -5, -5, -2, 4, 2, -2, -7, 3, 8, 5, -7, -6, 1, 14, 4, -9, -14, 2, 5, 5, -10, -7, 6, 22, 3, -12, -20, 1, 15, 15, -16, -12, 4, 25, 6, -14, -31, 13, 33, 14, -39, -32, -6, 39, 15, -20, -31, 33, 41, 14, -53, -44, 3, 66, 12, -35, -51, 22, 48, 36, -60, -43, 21

Offset: 0

Author

Stuart Clary, Apr 15 2006

Keywords

• sign
• easy

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..1000
• Peter Bala, Cyclotomic polynomials and the generating functions of A118205 and A118206
• Wikipedia, Cyclotomic polynomial
• Wikipedia, Liouville function

Crossrefs

Cf. A118205, A118207, A118208, A061020, A117209.

Programs

Mathematica

nmax = 100; lambda[k_Integer?Positive] := If[ k > 1, (-1)^Total[ Part[Transpose[FactorInteger[k]], 2] ], 1 ]; CoefficientList[ Series[ Product[ (1 - x^k)^(-lambda[k]), {k, 1, nmax} ], {x, 0, nmax} ], x ]
max = 100; s = Product[(1 - x^k)^(-LiouvilleLambda[k]), {k, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 23 2015 *)

PARI

{a(n)=polcoeff(exp(sum(m=1,n,sumdiv(m,d,d*moebius(core(d)))*x^m/m)+x*O(x^n)),n)} /* Cf. A061020 - Paul D. Hanna, Sep 22 2011 */

Formula

G.f.: A(x) = Product_{k>=1} (1 - x^k)^(-lambda(k)) where lambda(k) is the Liouville function, A008836.

Logarithmic derivative yields A061020. - Paul D. Hanna, Sep 22 2011

G.f.: A(x) = Product_{k >= 1} C(2*k,x^k), where C(k,x) denotes the k-th cyclotomic polynomial. - Peter Bala, Mar 31 2023