1,1

A polynomial is called flat iff it is of height 1, where the height is the largest absolute value of the coefficients.

A cyclotomic polynomial Phi[n] is said of order 3 iff n=pqr with distinct (usually odd) primes p,q,r.

It is well known that Phi[n] is flat if n has less than 3 odd prime factors, so this sequence includes all numbers of the form 2pq, with primes q>p>2, i.e. A075819. Sequence A117223 lists the complement, i.e. odd terms in this sequence, which start with 231 = 3*7*11.

Moreover, Kaplan shows that the present sequence also includes pqr if r = +-1 (mod pq). Sequence A160352 lists the subsequence of all such numbers, while A160354 lists elements which are not of this form.

Nathan Kaplan, Flat cyclotomic polynomials of order three, Journal of Number Theory 127 (2007) 118-126 (doi:10.1016/j.jnt.2007.01.008).

a(1)=30=2*3*5 is the smallest product of three distinct primes, and Phi[30] = X^8 + X^7 - X^5 - X^4 - X^3 + X + 1 has only coefficients in {0,1,-1}.

a(19)=231=3*7*11 is the smallest odd product of three distinct primes p,q,r such that Phi[pqr] is flat.

(PARI) for( pqr=1, 999, my(f=factor(pqr)); #f~==3 & vecmax(f[, 2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr", "))

Cf. A159908-A159909 (counts (p, q) for given r).

Sequence in context: A090815 A093599 A007304 this_sequence A053858 A075819 A034683

Adjacent sequences: A160347 A160348 A160349 this_sequence A160351 A160352 A160353

nonn

M. F. Hasler (MHasler(AT)univ-ag.fr), May 11 2009, May 14 2009