cp's OEIS Frontend

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

%I A320778 #6 Nov 21 2022 07:58:03
%S A320778 1,1,0,1,0,2,-3,4,-4,4,-9,14,-19,30,-42,50,-76,128,-194,286,-412,598,
%T A320778 -909,1386,-2100,3178,-4763,7122,-10758,16414,-25061,38056,-57643,
%U A320778 87568,-133436,203618,-311128,475536,-726355,1109718,-1697766,2601166,-3987903,6114666
%N A320778 Inverse Euler transform of the Euler totient function phi = A000010.
%C A320778 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 A320778 # The function EulerInvTransform is defined in A358451.
%p A320778 a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):
%p A320778 seq(a(n), n = 0..43); # _Peter Luschny_, Nov 21 2022
%t A320778 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 A320778 Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
%t A320778 EulerInvTransform[Array[EulerPhi,30]]
%Y A320778 Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
%Y A320778 Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
%Y A320778 Inverse Euler transforms: A059966, A320767, A320776, A320777, A320779, A320780, A320781, A320782.
%K A320778 sign
%O A320778 0,6
%A A320778 _Gus Wiseman_, Oct 22 2018