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%I A114536 #13 Jul 02 2018 06:59:17
%S A114536 1,1,1,1,1,2,1,1,1,2,1,3,1,2,3,1,1,2,1,4,3,2,1,3,1,2,1,4,1,12,1,1,3,2,
%T A114536 5,4,1,2,3,5,1,12,1,4,5,2,1,6,1,2,3,4,1,2,5,7,3,2,1,54,1,2,7,1,5,12,1,
%U A114536 4,3,32,1,8,1,2,3,4,7,12,1,7,1,2,1,55,5,2,3,8,1,58,7,4,3,2,5,6,1,2,9,4,1,12
%N A114536 Let the height of a polynomial be the largest coefficient in absolute value. Then a(n) is the maximal height of a divisor of x^n-1 with integral coefficients.
%H A114536 Antti Karttunen, <a href="/A114536/b114536.txt">Table of n, a(n) for n = 1..719</a>
%H A114536 Felipe Garcia H., <a href="http://fgarciah.bol.ucla.edu/Research/research.html">Research</a>.
%H A114536 Carl Pomerance and Nathan C. Ryan, <a href="http://projecteuclid.org/euclid.ijm/1258138432">The maximal height of divisors of x^n-1</a>, Illinois Journal of Mathematics 51 (2007) 597-604.
%H A114536 Nathan C. Ryan, <a href="http://www.math.ucla.edu/~nathan/research.html">Research</a>.
%F A114536 a(n)=1 iff n=1 or n=p^k where p is a prime and k is a positive integer; a(pq)=min{p,q} where p and q are distinct primes.
%e A114536 a(6)=2 since (x+1)(x^2+x+1)=x^3+2x^2+2x+1 divides x^6-1 and no other divisor has a greater height.
%t A114536 cyc[n_] := cyc[n] = Cyclotomic[n, x]; f[n_] := Block[{sd = Rest@ Subsets@ Divisors@ n, lst = {}, lmt = 2^DivisorSigma[0, n]}, For[i = 1, i < lmt, i++, AppendTo[lst, Max@ Abs@ CoefficientList[ Expand[ Times @@ (cyc[ # ] & /@ sd[[i]])], x]]]; Max@lst]; Array[f, 102] (* _Robert G. Wilson v_, Mar 01 2006 *)
%o A114536 (PARI) A114536(n) = { my(ds=divisors('x^n - 1),m=0); for(i=1,length(ds),for(j=0,poldegree(ds[i]),m = max(m,abs(polcoeff(ds[i],j))))); (m); }; \\ _Antti Karttunen_, Jul 01 2018
%o A114536 (PARI)
%o A114536 \\ This version needs less memory:
%o A114536 prod_by_bits(bits, fs) = { my(m=1,i=1); while(bits>0, if((bits%2),m *= fs[i]); i++; bits >>= 1); (m); };
%o A114536 A114536(n) = { my(fs=factor('x^n - 1)[,1],m=0,d); for(b=1,(2^#fs)-1,d = prod_by_bits(b,fs); for(j=0,poldegree(d),m = max(m,abs(polcoeff(d,j))))); (m); }; \\ _Antti Karttunen_, Jul 01 2018
%Y A114536 Cf. A117215 (number of divisors of x^n-1 having the maximal height).
%K A114536 nonn,nice
%O A114536 1,6
%A A114536 Felipe Garcia (fgarciah(AT)ucla.edu), Feb 15 2006
%E A114536 Edited and extended by _Robert G. Wilson v_, Mar 01 2006

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