A087653 Maximum difference between exponents in n-th cyclotomic polynomial.

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 2, 8, 1, 3, 1, 2, 2, 1, 1, 4, 5, 1, 9, 2, 1, 2, 1, 16, 2, 1, 4, 6, 1, 1, 2, 4, 1, 2, 1, 2, 6, 1, 1, 8, 7, 5, 2, 2, 1, 9, 4, 4, 2, 1, 1, 4, 1, 1, 6, 32, 4, 2, 1, 2, 2, 4, 1, 12, 1, 1, 10, 2, 6, 2, 1, 8, 27, 1, 1, 4, 4, 1, 2, 4, 1, 6, 6, 2, 2, 1, 4

Offset

1,4

Comments

Differs from A000190, A003557, A073752.

Links

Michel Marcus, Table of n, a(n) for n = 1..5000

Ala'a Al-Kateeb, Mary Ambrosino, Hoon Hong, and Eunjeong Lee, Maximum gap in cyclotomic polynomials, arXiv:1911.11667 [math.NT], 2019.

Hoon Hong, Eunjeong Lee, Hyang-Sook Lee, Cheol-Min Park, Maximum Gap in (Inverse) Cyclotomic Polynomial, arXiv:1101.4255 [math.NT], 2011.

Formula

a(p) = a(p*q) = p-1 for primes p < q (by Hong et al.). - Jonathan Sondow, Jan 09 2014

Example

Cyc(9) = x^6 + x^3 + x^0, so a(9) = 3.

Mathematica

a[n_] := Max[Differences[Exponent[Cyclotomic [n, x], x, List]]] (* Jonathan Sondow, Jan 09 2014 *)

Prog

(PARI) { mtermgap(pol)=local(p, m); m=0; p=0; for(k=0, poldegree(pol), if(polcoeff(pol, k)!=0, if(m<p, m=p); p=0, p=p+1)); max(m, p)+1 }
for(n=1, 200, print1(mtermgap(polcyclo(n))", "))

Crossrefs

Cf. A013595.

Sequence in context: A073752 A346487 A128708 * A295666 A355003 A322020

Adjacent sequences: A087650 A087651 A087652 * A087654 A087655 A087656

Keyword

nonn,easy

Author

Ralf Stephan, Sep 25 2003

Status

approved