A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

1, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, -1, -2, 1, 3, 1, -2, -2, 1, 0, -4, 0, 6, 6, -4, -8, 1, 4, -4, -5, 10, 16, -4, -25, -7, 17, 5, -16, 2, 42, 12, -58, -48, 40, 59, -27, -44, 67, 86, -103, -187, 36, 236, 45, -213, -5, 284, -23, -526, -188, 663, 520

Offset: 0

Author

Gus Wiseman, Oct 22 2018

Keywords

• sign

Comments

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Crossrefs

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.

Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.

Inverse Euler transforms: A059966, A320767, A320776, A320778, A320779, A320780, A320781, A320782.

Programs

Mathematica

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeNu,100]]