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

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

1,1

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.

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.

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A350352 A093599 A007304 * A053858 A075819 A306217

Adjacent sequences: A160347 A160348 A160349 * A160351 A160352 A160353

Keyword

nonn

Author

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

Status

approved