A086780 Number of negative terms in n-th cyclotomic polynomial.

1, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 3, 3, 0, 0, 1, 0, 2, 4, 5, 0, 1, 0, 6, 0, 3, 0, 3, 0, 0, 7, 8, 8, 1, 0, 9, 8, 2, 0, 4, 0, 5, 3, 11, 0, 1, 0, 2, 11, 6, 0, 1, 8, 3, 12, 14, 0, 3, 0, 15, 4, 0, 15, 7, 0, 8, 15, 8, 0, 1, 0, 18, 3, 9, 15, 8, 0, 2, 0, 20, 0, 4, 20, 21, 19, 5, 0, 3, 11, 11, 20, 23, 15, 1

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 03 2003

Keywords

• nonn

References

• See A051664

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A086765, A000961.

Cf. A051664 (number of nonzero terms in n-th cyclotomic polynomial).

Programs

Mathematica

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#<0&)], {n, 0, 100}]

PARI

a(n) = #select(x->(x<0), Vec(polcyclo(n))); \\ Michel Marcus, Apr 18 2018

Formula

a(n) = 0 iff n is a prime power. - T. D. Noe, Aug 08 2003

a(n) = (A051664(n)-1)/2 if n is not a prime power and has at most two distinct odd prime divisors. So 105 is the smallest n>1 where neither formula applies. - Aaron Meyerowitz, Apr 18 2018

Extensions

More terms from T. D. Noe, Aug 08 2003