Search: id:A160350
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%I A160350
%S A160350 30,42,66,70,78,102,110,114,130,138,154,170,174,182,186,190,222,230,231,
%T A160350 238,246,258,266,282,286,290,310,318,322,354,366,370,374,399,402,406,
%U A160350 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.
%D A160350 Nathan Kaplan, Flat cyclotomic polynomials of order three, Journal of 
               Number Theory 127 (2007) 118-126 (doi:10.1016/j.jnt.2007.01.008).
%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).
%Y A160350 Sequence in context: A090815 A093599 A007304 this_sequence A053858 A075819 
               A034683
%Y A160350 Adjacent sequences: A160347 A160348 A160349 this_sequence A160351 A160352 
               A160353
%K A160350 nonn
%O A160350 1,1
%A A160350 M. F. Hasler (MHasler(AT)univ-ag.fr), May 11 2009, May 14 2009

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