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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. 7
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

LINKS

Antti Karttunen, Table of n, a(n) for n = 1..719

Felipe Garcia H., Research.

Carl Pomerance and Nathan C. Ryan, The maximal height of divisors of x^n-1, Illinois Journal of Mathematics 51 (2007) 597-604.

Nathan C. Ryan, Research.

FORMULA

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.

EXAMPLE

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.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A117215 (number of divisors of x^n-1 having the maximal height).

Sequence in context: A085091 A052128 A284600 * A330692 A138010 A206487

Adjacent sequences:  A114533 A114534 A114535 * A114537 A114538 A114539

KEYWORD

nonn,nice

AUTHOR

Felipe Garcia (fgarciah(AT)ucla.edu), Feb 15 2006

EXTENSIONS

Edited and extended by Robert G. Wilson v, Mar 01 2006

STATUS

approved

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Last modified April 11 22:01 EDT 2021. Contains 342888 sequences. (Running on oeis4.)