[seqfan] count of terms in cyclotomic polynomial
wouter meeussen
wouter.meeussen at pandora.be
Sun Jan 18 14:20:20 CET 2009
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 *)
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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]
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