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Maximum difference between exponents in n-th cyclotomic polynomial.
2

%I #22 Nov 27 2019 11:16:13

%S 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,

%T 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,

%U 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

%N Maximum difference between exponents in n-th cyclotomic polynomial.

%C Differs from A000190, A003557, A073752.

%H Michel Marcus, <a href="/A087653/b087653.txt">Table of n, a(n) for n = 1..5000</a>

%H Ala'a Al-Kateeb, Mary Ambrosino, Hoon Hong, and Eunjeong Lee, <a href="https://arxiv.org/abs/1911.11667">Maximum gap in cyclotomic polynomials</a>, arXiv:1911.11667 [math.NT], 2019.

%H Hoon Hong, Eunjeong Lee, Hyang-Sook Lee, Cheol-Min Park, <a href="http://arxiv.org/abs/1101.4255">Maximum Gap in (Inverse) Cyclotomic Polynomial</a>, arXiv:1101.4255 [math.NT], 2011.

%F a(p) = a(p*q) = p-1 for primes p < q (by Hong et al.). - _Jonathan Sondow_, Jan 09 2014

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

%t a[n_] := Max[Differences[Exponent[Cyclotomic [n, x], x, List]]] (* _Jonathan Sondow_, Jan 09 2014 *)

%o (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 }

%o for(n=1, 200, print1(mtermgap(polcyclo(n))", "))

%Y Cf. A013595.

%K nonn,easy

%O 1,4

%A _Ralf Stephan_, Sep 25 2003