

A117223


Numbers n such that Phi(n,x) is a flat cyclotomic polynomial of order three.


19



231, 399, 435, 465, 483, 651, 663, 741, 861, 885, 903, 915, 1113, 1173, 1209, 1281, 1311, 1335, 1353, 1443, 1479, 1533, 1581, 1599, 1653, 1743, 1833, 1947, 2163, 2211, 2235, 2247, 2265, 2301, 2337, 2379, 2409, 2485, 2667, 2685, 2715, 2829, 2877, 2915
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OFFSET

1,1


COMMENTS

A flat polynomial is defined to be a polynomial whose coefficients are 1, 0, or 1. Order three means that n is the product of three odd primes p < q < r. Bachman shows that for each p there are an infinite number of pairs {q,r} that generate flat cyclotomic polynomials. It is well known that all cyclotomic polynomials of orders one and two are flat. There are no flat cyclotomic polynomials of order four for n < 10^5.
Kaplan shows that the sequence also includes pqr if r = +1 (mod pq). Sequence A160353 lists the subsequence of all odd numbers of this form, while A160355 lists the elements which are not of this form. More cases are covered by David Broadhurst's conjectures, cf. link.  M. F. Hasler, May 15 2009


LINKS



FORMULA



MATHEMATICA

IsOrder3[n_] := (n>1) && OddQ[n] && Transpose[FactorInteger[n]][[2]] == {1, 1, 1}; PolyHeight[p_] := Max[Abs[CoefficientList[p, x]]]; Clear[x]; Select[Range[4000], IsOrder3[ # ] && PolyHeight[Cyclotomic[ #, x]]==1&]


PROG

(PARI) A117223(n, show=0)={ my(pqr=1, f); while(n, matsize(f=factor(pqr+=2))[1]==3 & vecmax(f[, 2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & n & show & print1(pqr", ")); pqr } \\ M. F. Hasler, May 15 2009


CROSSREFS

Cf. A117318 (fourthorder flat cyclotomic polynomials).


KEYWORD

nonn


AUTHOR



STATUS

approved



