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

30, 42, 66, 70, 78, 102, 110, 114, 130, 138, 154, 170, 174, 182, 186, 190, 222, 230, 231, 238, 246, 258, 266, 282, 286, 290, 310, 318, 322, 354, 366, 370, 374, 399, 402, 406, 410, 418, 426, 430, 434, 435, 438, 442, 465, 470, 474, 483, 494, 498, 506, 518, 530

Offset: 1

Author

M. F. Hasler, May 11 2009, May 14 2009

Keywords

• nonn

Comments

A polynomial is called flat iff it is of height 1, where the height is the largest absolute value of the coefficients.

A cyclotomic polynomial phi(n) is said of order 3 iff n=pqr with distinct (usually odd) primes p,q,r.

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.

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.

Examples

			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}.
a(19)=231=3*7*11 is the smallest odd product of three distinct primes p,q,r such that Phi[pqr] is flat.
		

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

Crossrefs

Cf. A159908, A159909 (counts (p, q) for given r).

Programs

PARI

for( pqr=1,999, my(f=factor(pqr)); #f~==3 & vecmax(f[,2])==1 & vecmax(abs(Vec(polcyclo(pqr))))==1 & print1(pqr","))