A159908 Number of pairs (p,q) of primes p <= q <= r=prime(n) such that the cyclotomic polynomial Phi(p*q*r) has no coefficient > 1 in absolute value.

1, 3, 6, 9, 13, 15, 19, 23, 27, 30, 34, 35, 43, 40, 45, 47, 54, 57, 58, 64, 69, 69, 71, 79, 79, 84, 86, 87, 97, 96, 94, 107, 106, 109, 120, 111, 120, 123, 124, 133, 135, 134, 144, 143, 143, 154, 154, 154, 163, 161, 167, 175, 174, 175, 179, 183, 187, 191, 193, 199, 197, 202, 203

Offset: 1

Author

M. F. Hasler, May 09 2009

Keywords

• hard
• nonn

Comments

The cyclotomic polynomial Phi[pqr] (p,q,r primes) can only have coefficients with absolute value > 1 if p,q,r are distinct odd primes. This sequence also counts the trivial cases where (1): p=2, or (2): p=q, or (3): q=r. The number of these cases is A008486(n-1). Sequence A159909 counts only the nontrivial cases.

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. A008486, A159909.

Programs

PARI

A159908(n) = sum( i=1,n, my(pq=prime(n)*prime(i)); sum( j=1,i, vecmax(abs(Vec(polcyclo(prime(j)*pq))))==1 ))

Formula

a(n) = A008486(n-1) + A159909(n).

Extensions

More terms from Robin Visser, Aug 09 2023