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

1,2

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.

Formula

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

Prog

(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 ))

Crossrefs

Cf. A008486, A159909.

Sequence in context: A078559 A317557 A338763 * A236761 A088364 A022853

Adjacent sequences: A159905 A159906 A159907 * A159909 A159910 A159911

Keyword

hard,nonn

Author

M. F. Hasler, May 09 2009

Extensions

More terms from Robin Visser, Aug 09 2023

Status

approved