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A114536
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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.
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7
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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, 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, 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
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OFFSET
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1,6
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LINKS
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FORMULA
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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.
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
(PARI)
\\ This version needs less memory:
prod_by_bits(bits, fs) = { my(m=1, i=1); while(bits>0, if((bits%2), m *= fs[i]); i++; bits >>= 1); (m); };
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
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CROSSREFS
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Cf. A117215 (number of divisors of x^n-1 having the maximal height).
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KEYWORD
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nonn,nice
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AUTHOR
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Felipe Garcia (fgarciah(AT)ucla.edu), Feb 15 2006
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EXTENSIONS
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STATUS
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approved
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