cp's OEIS Frontend

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

%I A320776 #10 Nov 21 2022 08:18:26
%S A320776 1,0,1,1,1,0,-1,-1,0,1,0,-1,-1,-1,1,3,3,-2,-5,-4,0,7,7,0,-9,-10,2,15,
%T A320776 15,-3,-27,-30,3,46,51,1,-71,-91,-7,117,157,23,-194,-265,-57,318,465,
%U A320776 111,-536,-821,-230,893,1456,505,-1485,-2559,-1036,2433,4483,2022
%N A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.
%C A320776 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.
%p A320776 # The function EulerInvTransform is defined in A358451.
%p A320776 a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-NumberOfPrimeFactors(n))):
%p A320776 seq(a(n), n = 0..59); # _Peter Luschny_, Nov 21 2022
%t A320776 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}]]];
%t A320776 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
%t A320776 EulerInvTransform[Array[PrimeOmega,100]]
%Y A320776 Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
%Y A320776 Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
%Y A320776 Inverse Euler transforms: A059966, A320767, A320777, A320778, A320779, A320780, A320781, A320782.
%K A320776 sign
%O A320776 0,16
%A A320776 _Gus Wiseman_, Oct 22 2018