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A160495 Irregular triangle of residue classes (mod pq) of primes r such that the cyclotomic polynomial Phi(pqr,x) is flat.

%I A160495 #10 Apr 14 2017 03:00:21
%S A160495 1,14,1,2,10,11,19,20,1,7,8,25,26,32,1,34,1,2,8,17,19,20,22,31,37,38,
%T A160495 1,13,20,22,23,28,29,31,38,50,1,2,53,54,1,2,7,13,16,23,28,29,34,41,44,
%U A160495 50,55,56,1,64,1,7,8,10,17,19,28,41,50,52,59,61,62,68
%N A160495 Irregular triangle of residue classes (mod pq) of primes r such that the cyclotomic polynomial Phi(pqr,x) is flat.
%C A160495 A polynomial is flat if its coefficients are 1, 0, or -1. The values of pq are in sequence A046388. Each row begins with 1 and ends with pq-1. For each number k in a row, the number pq-k is also in the row. Row n has 2*A160496(n) terms. For the pq in sequence A160497, the row has only two terms. By Kaplan's theorems 2 and 3, only the first prime r in each residue class 1..(p-1)(q-1)/2 needs to be checked to determine whether the residue class produces flat cyclotomic polynomials. Is there a simplier method of finding these residue classes?
%H A160495 T. D. Noe, <a href="/A160495/b160495.txt">Rows n=1..100 of triangle, flattened</a>
%H A160495 Nathan Kaplan, <a href="https://doi.org/10.1016/j.jnt.2007.01.008">Flat cyclotomic polynomials of order three</a>, J. Number Theory 127 (2007), 118-126.
%e A160495 The second row (1,2,10,11,19,20) is for pq=21. If r is a prime with r mod pq equal to one of these 6 values, then Phi(21*r,x) is flat.
%Y A160495 Cf. A046388, A117223.
%K A160495 nonn,tabf
%O A160495 1,2
%A A160495 _T. D. Noe_, May 15 2009