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A160354 Indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq). 3

%I #14 Aug 02 2023 08:06:55

%S 70,130,154,170,230,231,238,266,286,322,370,374,399,418,430,434,442,

%T 470,483,494,518,530,598,638,646,651,658,663,670,682,730,741,742,754,

%U 782,806,814,826,830,854,874,902,938,962,970,986,1022,1030,1034,1054,1066

%N Indices pqr of flat cyclotomic polynomials of order 3 which are not of the form r = +/-1 (mod pq).

%C Kaplan (2007) has shown that Phi(pqr) has coefficients in {0,1,-1} if r = +-1 (mod pq), where p<q<r are primes. Here we list the elements of A160350 which do not satisfy this equality.

%C Yet most elements are even, i.e. in A075819. Sequence A160355 is the subsequence of odd terms. See A160350 for more details.

%H Robin Visser, <a href="/A160354/b160354.txt">Table of n, a(n) for n = 1..10000</a>

%H 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.

%F Equals A160350 \ A160352.

%e a(1)=70=2*5*7 is the smallest element of A160350 for which the largest factor (7) is not congruent to +- 1 modulo the product of the smaller factors (2*5).

%o (PARI) for( pqr=1,1999, my(f=factor(pqr)); #f~==3 & vecmax(f[,2])==1 & abs((f[3,1]+1)%(f[1,1]*f[2,1])-1)!=1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr","))

%K nonn

%O 1,1

%A _M. F. Hasler_, May 11 2009

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