A117223 Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order three.

231, 399, 435, 465, 483, 651, 663, 741, 861, 885, 903, 915, 1113, 1173, 1209, 1281, 1311, 1335, 1353, 1443, 1479, 1533, 1581, 1599, 1653, 1743, 1833, 1947, 2163, 2211, 2235, 2247, 2265, 2301, 2337, 2379, 2409, 2485, 2667, 2685, 2715, 2829, 2877, 2915

Offset: 1

Author

T. D. Noe, Mar 04 2006

Keywords

• nonn

Comments

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 David Broadhurst's conjectures, cf. link. - M. F. Hasler, May 15 2009

Links

• David Broadhurst and T. D. Noe, Table of n, a(n) for n = 1..10000
• Gennady Bachman, Flat cyclotomic polynomials of order three, Bull. London Math. Soc. 38 (2006), 53-60.
• David Broadhurst, Flat ternary cyclotomic polynomials, in: Yahoo! group "primenumbers", May 2009. [Broken link]
• Phil Carmody and others, Cyclotomic polynomial puzzles, digest of 43 messages in primenumbers Yahoo group, May 9 - May 23, 2009.
• Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.
• Carlo Sanna, A Survey on Coefficients of Cyclotomic Polynomials, arXiv:2111.04034 [math.NT], 2021.

Crossrefs

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

Programs

Mathematica

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 } \\ M. F. Hasler, May 15 2009

Formula

Equals A160353 union A160355 = A160350 \ A075819 = A160350 intersect A046389. - M. F. Hasler, May 15 2009