A086798 Number of coefficients equal to zero in n-th cyclotomic polynomial.

0, 0, 0, 1, 0, 0, 0, 3, 4, 0, 0, 2, 0, 0, 2, 7, 0, 4, 0, 4, 4, 0, 0, 6, 16, 0, 16, 6, 0, 2, 0, 15, 6, 0, 8, 10, 0, 0, 8, 12, 0, 4, 0, 10, 18, 0, 0, 14, 36, 16, 10, 12, 0, 16, 24, 18, 12, 0, 0, 10, 0, 0, 28, 31, 18, 6, 0, 16, 14, 8, 0, 22, 0, 0, 34, 18, 30, 8, 0, 28, 52, 0, 0, 16, 24, 0, 18, 30, 0

Offset: 1

Author

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 05 2003

Keywords

• nonn

References

• See A051664.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Crossrefs

Cf. A086765, A086780.

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)=sum(k=0,eulerphi(n),if(polcoeff(polcyclo(n),k),0,1))

PARI

A086798(n) = (1 + eulerphi(n) - length(select(x->x!=0, Vec(polcyclo(n))))); \\ Antti Karttunen, Sep 21 2018

Formula

From Benoit Cloitre, Aug 06 2003: (Start)

a(4n+2) = a(2n+1); a(4n) = a(2n) + phi(2n).

When p is an odd prime and m integer >= 1: a(p^m) = a(2*p^m) = p^m - p^(m-1) - p + 1. In particular a(p) = a(2p) = 0. (End)

a(n) = 1 + phi(n) - A051664(n). - T. D. Noe, Aug 08 2003

Extensions

More terms from Benoit Cloitre and T. D. Noe, Aug 06 2003