the count of terms in cyclotomic polynomial bigPhi(n,x), say c, and the largest prime factor of n, say p, are known to coïncide for all n except those in a beheaded A070537= [[ 1 ]] ,15,21,30,33,35,39,42,45... in which case p<c (conjecture) It seems that A070537 also equals the values of n where w(n) is *larger* than the largest prime factor of n. with w(n)=1+ InverseMoebiusTransform( seq_z ) with seq_z equal to the sequence defined by mu(n) Sum(d|n, phi(d) mu(d) ) we get p<c<w OR p=c=w I don't see an evident link, do you? Wouter. (* sequences seq_z and w(n) pending desimbicilisation *) ____________________________________________ Please ignore the Mathematica code below: w=1+invmoebius[Table[MoebiusMu[n] Fold[ EulerPhi[#2] MoebiusMu[#2]+#1&, 0, Divisors[n]],{n,100}]] and invmoebT[partiallist:{__Integer}]:=Block[{n=Length[partiallist]},Fold[#1+par tiallist[[#2]]&,0,Divisors[n]]]; invmoebius[argSeq:{__Integer}]:=Table[invmoebT[Take[argSeq,i]],{i,Length[arg Seq]}]; c=Table[Length[Cyclotomic[n,x]], {n,100}] p=Table[If[n===1,2,Part[FactorInteger[n],-1,1]], {n,100}] triad=Transpose[{c,p,w}] diff=Select[triad,UnsameQ@@#&] A070537=Flatten@ Position[triad,q_List/;UnsameQ @@ q] ____________________________________________