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A160350 Indices n=pqr of flat cyclotomic polynomials, where p are primes.

%I A160350 #14 Aug 02 2023 08:06:49
%S A160350 30,42,66,70,78,102,110,114,130,138,154,170,174,182,186,190,222,230,
%T A160350 231,238,246,258,266,282,286,290,310,318,322,354,366,370,374,399,402,
%U A160350 406,410,418,426,430,434,435,438,442,465,470,474,483,494,498,506,518,530
%N A160350 Indices n=pqr of flat cyclotomic polynomials, where p<q<r are primes.
%C A160350 A polynomial is called flat iff it is of height 1, where the height is the largest absolute value of the coefficients.
%C A160350 A cyclotomic polynomial phi(n) is said of order 3 iff n=pqr with distinct (usually odd) primes p,q,r.
%C A160350 It is well known that phi(n) is flat if n has less than 3 odd prime factors, so this sequence includes all numbers of the form 2pq, with primes q>p>2, i.e. A075819. Sequence A117223 lists the complement, i.e. odd terms in this sequence, which start with 231 = 3*7*11.
%C A160350 Moreover, Kaplan shows that the present sequence also includes pqr if r = +-1 (mod pq). Sequence A160352 lists the subsequence of all such numbers, while A160354 lists elements which are not of this form.
%H A160350 Robin Visser, <a href="/A160350/b160350.txt">Table of n, a(n) for n = 1..10000</a>
%H A160350 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 A160350 a(1)=30=2*3*5 is the smallest product of three distinct primes, and Phi[30] = X^8 + X^7 - X^5 - X^4 - X^3 + X + 1 has only coefficients in {0,1,-1}.
%e A160350 a(19)=231=3*7*11 is the smallest odd product of three distinct primes p,q,r such that Phi[pqr] is flat.
%o A160350 (PARI) for( pqr=1,999, my(f=factor(pqr)); #f~==3 & vecmax(f[,2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr","))
%Y A160350 Cf. A159908, A159909 (counts (p, q) for given r).
%K A160350 nonn
%O A160350 1,1
%A A160350 _M. F. Hasler_, May 11 2009, May 14 2009