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A146961 Numbers k = p*q*r, with odd primes p < q < r, such that Sister Beiter's cyclotomic coefficient conjecture is false.

%I A146961 #23 Aug 25 2023 12:42:09
%S A146961 20213,125609,136477,141317,150271,198493,199177,212971,239039,273229,
%T A146961 282367,291343,311201,332777,373901,393313,398563,412357,442091,
%U A146961 449527,449647,450131,456569,461263,469249,470741,475057,522461,524837,532363
%N A146961 Numbers k = p*q*r, with odd primes p < q < r, such that Sister Beiter's cyclotomic coefficient conjecture is false.
%C A146961 In 1968, Sister Beiter conjectured that for k = p*q*r, with odd primes p < q < r, the maximum coefficient (in absolute value) of the cyclotomic polynomial Phi(k,x) is <= (p+1)/2. Up to 10^6, all counterexamples have p > 7. Gallot and Moree prove the conjecture is false for p > 7.
%H A146961 Robin Visser, <a href="/A146961/b146961.txt">Table of n, a(n) for n = 1..200</a>
%H A146961 A. S. Bang, <a href="https://www.jstor.org/stable/24526469">Om Ligningen phi_n(x) = 0</a>, Nyt tidsskrift for matematik, Vol. 6, Afdeling B (1895), pp. 6-12 (7 pages).
%H A146961 Yves Gallot and Pieter Moree, <a href="https://archive.mpim-bonn.mpg.de/id/eprint/410/">Counter-examples to Sister Beiter's cyclotomic coefficient conjecture</a>, MPIM Preprint Series 2007 (141).
%H A146961 Nathan Kaplan, <a href="https://doi.org/10.1016/j.jnt.2007.01.008">Flat cyclotomic polynomials of order three</a>, Journal of Number Theory, Volume 127, Issue 1, November 2007, Pages 118-126.
%H A146961 G. S. Kazandzidis, <a href="http://www.hms.gr/apothema/?s=sap&amp;i=50">On the cyclotomic polynomial: Coefficients</a>, Bull. Soc. Math. Gr`ece (N.S.) 4 (1963), no. 1, 1-11.
%H A146961 Carlo Sanna, <a href="https://arxiv.org/abs/2111.04034">A Survey on Coefficients of Cyclotomic Polynomials</a>, arXiv:2111.04034 [math.NT], 2021.
%H A146961 Wikipedia, <a href="https://en.wikipedia.org/wiki/Marion_Beiter">Marion Beiter</a>.
%o A146961 (PARI) isok(m) = if ((m%2) && (bigomega(m)==3) && (omega(m)==3), my(p=vecmin(factor(m)[,1])); vecmax(apply(abs, Vec(polcyclo(m)))) > (p+1)/2;); \\ _Michel Marcus_, Jan 16 2023
%o A146961 (Sage)
%o A146961 from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs
%o A146961 for n in range(3, 100000, 2):
%o A146961     pqr = Integer(n).prime_factors()
%o A146961     if (len(pqr) == 3) and (product(pqr) == n):
%o A146961         coeffs = cyclotomic_coeffs(n, sparse=False)
%o A146961         max_coeff = max(abs(c) for c in coeffs)
%o A146961         if (max_coeff > (pqr[0]+1)//2): print(n)  # _Robin Visser_, Aug 17 2023
%Y A146961 Subsequence of A046389.
%K A146961 nonn
%O A146961 1,1
%A A146961 _T. D. Noe_, Nov 03 2008