login
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.
3
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
1,8
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.
LINKS
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
EXAMPLE
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).
PROG
(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 ))
CROSSREFS
Cf. A117223. [T. D. Noe, May 11 2009]
Sequence in context: A106448 A295280 A307838 * A176004 A214738 A353360
KEYWORD
nonn,hard
AUTHOR
M. F. Hasler, May 09 2009
EXTENSIONS
Extended by T. D. Noe, May 11 2009
More terms from Robin Visser, Aug 09 2023
STATUS
approved