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A160495 Irregular triangle of residue classes (mod pq) of primes r such that the cyclotomic polynomial Phi(pqr,x) is flat. 4
1, 14, 1, 2, 10, 11, 19, 20, 1, 7, 8, 25, 26, 32, 1, 34, 1, 2, 8, 17, 19, 20, 22, 31, 37, 38, 1, 13, 20, 22, 23, 28, 29, 31, 38, 50, 1, 2, 53, 54, 1, 2, 7, 13, 16, 23, 28, 29, 34, 41, 44, 50, 55, 56, 1, 64, 1, 7, 8, 10, 17, 19, 28, 41, 50, 52, 59, 61, 62, 68 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A polynomial is flat if its coefficients are 1, 0, or -1. The values of pq are in sequence A046388. Each row begins with 1 and ends with pq-1. For each number k in a row, the number pq-k is also in the row. Row n has 2*A160496(n) terms. For the pq in sequence A160497, the row has only two terms. By Kaplan's theorems 2 and 3, only the first prime r in each residue class 1..(p-1)(q-1)/2 needs to be checked to determine whether the residue class produces flat cyclotomic polynomials. Is there a simplier method of finding these residue classes?

LINKS

T. D. Noe, Rows n=1..100 of triangle, flattened

Nathan Kaplan, Flat cyclotomic polynomials of order three, J. Number Theory 127 (2007), 118-126.

EXAMPLE

The second row (1,2,10,11,19,20) is for pq=21. If r is a prime with r mod pq equal to one of these 6 values, then Phi(21*r,x) is flat.

CROSSREFS

Cf. A046388, A117223.

Sequence in context: A209601 A040201 A179948 * A040204 A040203 A040205

Adjacent sequences:  A160492 A160493 A160494 * A160496 A160497 A160498

KEYWORD

nonn,tabf

AUTHOR

T. D. Noe, May 15 2009

STATUS

approved

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Last modified March 23 18:13 EDT 2019. Contains 321433 sequences. (Running on oeis4.)