1,1

A flat polynomial is defined to be a polynomial whose coefficients are -1, 0, or 1. Order three means that n is the product of three odd primes p<q<r. Bachman shows that for each p there are an infinite number of pairs {q,r} that generate flat cyclotomic polynomials. It is well known that all cyclotomic polynomials of orders one and two are flat. There are no flat cyclotomic polynomials of order four for n<10^5.

Kaplan shows that the sequence also includes pqr if r = +-1 (mod pq). Sequence A160353 lists the subsequence of all odd numbers of this form, while A160355 lists the elements which are not of this form. More cases are covered by D. Broadhurst's conjectures, cf. link. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 15 2009]

Gennady Bachman, Flat cyclotomic polynomials of order three, Bull. London Math. Soc. 38 (2006), 53-60.

Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

David Broadhurst and T. D. Noe, Table of n, a(n) for n=1..10000

D. Broadhurst, Flat ternary cyclotomic polynomials, in: primenumbers group, May 2009. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 15 2009]

A117223 = A160353 union A160355 = A160350 \ A075819 = A160350 intersect A046389. [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 15 2009]

IsOrder3[n_] := (n>1) && OddQ[n] && Transpose[FactorInteger[n]][[2]] == {1, 1, 1}; PolyHeight[p_] := Max[Abs[CoefficientList[p, x]]]; Clear[x]; Select[Range[4000], IsOrder3[ # ] && PolyHeight[Cyclotomic[ #, x]]==1&]

(PARI) A117223(n, show=0)={ my(pqr=1, f); while(n, matsize(f=factor(pqr+=2))[1]==3 & vecmax(f[, 2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & n-- & show & print1(pqr", ")); pqr } [From M. F. Hasler (MHasler(AT)univ-ag.fr), May 15 2009]

Cf. A117318 (fourth-order flat cyclotomic polynomials).

Sequence in context: A031965 A088289 A046009 this_sequence A160355 A029569 A152102

Adjacent sequences: A117220 A117221 A117222 this_sequence A117224 A117225 A117226

nonn

T. D. Noe (noe(AT)sspectra.com), Mar 04 2006