This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A117223 #36 Aug 06 2022 07:23:12
%S A117223 231,399,435,465,483,651,663,741,861,885,903,915,1113,1173,1209,1281,
%T A117223 1311,1335,1353,1443,1479,1533,1581,1599,1653,1743,1833,1947,2163,
%U A117223 2211,2235,2247,2265,2301,2337,2379,2409,2485,2667,2685,2715,2829,2877,2915
%N A117223 Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order three.
%C A117223 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.
%C A117223 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
%H A117223 David Broadhurst and T. D. Noe, <a href="/A117223/b117223.txt">Table of n, a(n) for n = 1..10000</a>
%H A117223 Gennady Bachman, <a href="http://dx.doi.org/10.1112/S0024609305018096">Flat cyclotomic polynomials of order three</a>, Bull. London Math. Soc. 38 (2006), 53-60.
%H A117223 David Broadhurst, <a href="http://groups.yahoo.com/group/primenumbers/message/20305">Flat ternary cyclotomic polynomials, in: Yahoo! group "primenumbers"</a>, May 2009. [Broken link]
%H A117223 Phil Carmody and others, <a href="/A117223/a117223.txt">Cyclotomic polynomial puzzles</a>, digest of 43 messages in primenumbers Yahoo group, May 9 - May 23, 2009.
%H A117223 Nathan Kaplan, <a href="http://dx.doi.org/10.1016/j.jnt.2007.01.008">Flat cyclotomic polynomials of order three</a>, J. Number Theory 127 (2007), 118-126.
%H A117223 Carlo Sanna, <a href="https://arxiv.org/abs/2111.04034">A Survey on Coefficients of Cyclotomic Polynomials</a>, arXiv:2111.04034 [math.NT], 2021.
%F A117223 Equals A160353 union A160355 = A160350 \ A075819 = A160350 intersect A046389. - _M. F. Hasler_, May 15 2009
%t A117223 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&]
%o A117223 (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
%Y A117223 Cf. A117318 (fourth-order flat cyclotomic polynomials).
%K A117223 nonn
%O A117223 1,1
%A A117223 _T. D. Noe_, Mar 04 2006