Search: id:A138475
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%I A138475
%S A138475 0,1,3,5,5,7,7,105,11,11,11,385,13,429,715,715,165,323323,15015,323323,
%T A138475 1062347,1062347,373065,1062347,11305,1062347,1062347,1062347,37182145,
%U A138475 2800733,37182145,5107219,40755,40755,275147873,10015005,215656441
%N A138475 Least k such that the x^n coefficient of cyclotomic polynomial Phi(k,
               x) has the largest possible magnitude.
%C A138475 The maximum possible magnitude of the x^n coefficient is A138474(n). 
               Note that a(0)=0 because we assume Phi(0,x)=1; another convention 
               has Phi(0,x)=x, which would force a(0) and a(1) to be reversed.
%C A138475 It appears that (1) for n>80, a(n) has an even number of prime factors 
               and (2) for prime n>80, n divides a(n). Terms up to n=128 were found 
               by exhaustive search; subsequent terms were found by a much faster 
               hill-climbing method.
%D A138475 See A138474
%H A138475 T. D. Noe, <a href="b138475.txt">Table of n, a(n) for n=0..1000</a>
%e A138475 a(7)=105 because the cyclotomic polynomial Phi(105,x) has the term -2x^7.
%t A138475 coef[k_,n_] := Module[{t, b=Table[0,{k+1}]}, t=-MoebiusMu[n]*Table[g=GCD[n,
               k-m]; MoebiusMu[g]*EulerPhi[g], {m,0,k-1}]; b[[1]]=1; Do[b[[j+1]] 
               = Take[b,j].Take[t,-j]/j, {j,k}]; b]; Table[mx=1; r=PrimePi[k]+1; 
               mnN=Prime[r]; ps=Reverse[Prime[Range[r]]]; Do[d=IntegerDigits[i,2,
               r]; n=Times@@Pick[ps,d,1]; c=Abs[coef[k,n][[ -1]]]; If[c==mx, mnN=Min[mnN,
               n], If[c>mx, mx=c; mnN=n]], {i,2^r-1}]; mnN, {k,2,20}]
%Y A138475 Cf. A013594 (smallest order of cyclotomic polynomial containing n or 
               -n as a coefficient).
%Y A138475 Sequence in context: A087821 A109258 A088081 this_sequence A023840 A131421 
               A088743
%Y A138475 Adjacent sequences: A138472 A138473 A138474 this_sequence A138476 A138477 
               A138478
%K A138475 nonn
%O A138475 0,3
%A A138475 T. D. Noe (noe(AT)sspectra.com), Mar 19 2008, Apr 14 2008, Feb 16 2009

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