A159909 Number of pairs (p,q) of odd primes p < q < r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.

0, 0, 0, 0, 1, 0, 1, 2, 3, 3, 4, 2, 7, 1, 3, 2, 6, 6, 4, 7, 9, 6, 5, 10, 7, 9, 8, 6, 13, 9, 4, 14, 10, 10, 18, 6, 12, 12, 10, 16, 15, 11, 18, 14, 11, 19, 16, 13, 19, 14, 17, 22, 18, 16, 17, 18, 19, 20, 19, 22, 17, 19, 17, 19, 19, 19, 31, 25, 13, 38, 20, 23, 25, 23, 31, 30, 31, 19

Offset: 1

Author

M. F. Hasler, May 09 2009

Keywords

• nonn
• hard

Comments

The cyclotomic polynomial Phi[pqr] can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes, that's why we require 2 < p < q < r. If any of these inequalities is replaced by equality, then Phi[pqr] necessarily has only zero or unit (+-1) coefficients. Sequence A159908 counts all possibilities including these trivial cases.

Examples

			a(5)=1 is the first nonzero term, since the smallest example for Phi(pqr) having no coefficient > 1 (in abs. value) for odd primes p<q<r is obtained for r=prime(5), namely Phi(3*7*11).
		

Links

• Robin Visser, Table of n, a(n) for n = 1..130
• Phil Carmody, "Cyclotomic polynomial puzzles", in: "primenumbers" group, May 9, 2009.
• Phil Carmody, David Broadhurst, Maximilian Hasler, Makoto Kamada, Cyclotomic polynomial puzzles, digest of 43 messages in primenumbers Yahoo group, May 9, 2009 - May 23, 2013.
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Crossrefs

Cf. A117223. [T. D. Noe, May 11 2009]

Programs

PARI

A159909(n) = sum( i=2,n-1, my(pq=prime(n)*prime(i)); sum( j=2,i-1, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))

Extensions

Extended by T. D. Noe, May 11 2009

More terms from Robin Visser, Aug 09 2023