A051664 a(n) is the number of nonzero coefficients in the n-th cyclotomic polynomial.

2, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 7, 2, 17, 3, 19, 5, 9, 11, 23, 3, 5, 13, 3, 7, 29, 7, 31, 2, 15, 17, 17, 3, 37, 19, 17, 5, 41, 9, 43, 11, 7, 23, 47, 3, 7, 5, 23, 13, 53, 3, 17, 7, 25, 29, 59, 7, 61, 31, 9, 2, 31, 15, 67, 17, 31, 17, 71, 3, 73, 37, 7, 19, 31, 17, 79, 5, 3

Offset

1,1

Comments

a(n)=p(n) if n=p(n); a(n) is not always A006530(n). - Labos Elemer, May 03 2002

This sequence is the Mobius transform of A087073. Let m be the squarefree part of n, then a(n) = a(m). When n = pq, the product of two distinct odd primes, then there is a formula for a(pq). Let x = 1/p (mod q) and y = 1/q (mod p). Then a(pq) = 2xy-1. There are also formulas for the number of positive and negative terms. See papers by Carlitz or Lam and Leung. - T. D. Noe, Aug 08 2003

Links

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

L. Carlitz, Number of terms in the cyclotomic polynomial F(pq,x), Amer. Math. Monthly, Vol. 73, No. 9, 1966, pp. 979-981.

T. Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi(pq,x), Amer. Math. Monthly, Vol. 103, No. 7, 1996, pp. 562-564.

Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Formula

a(n) = phi(n) + 1 - A086798(n). - T. D. Noe, Aug 08 2003

Example

9th cyclotomic polynomial is x^6+x^3+1 which has 3 terms, so a(9)=3.

Maple

A051664 := proc(n)
        numtheory[cyclotomic](n, x) ;
        nops([coeffs(%)]) ;
end proc: # R. J. Mathar, Sep 15 2012

Mathematica

Table[Count[CoefficientList[Cyclotomic[n, x], x], _?(#!=0&)], {n, 0, 100}]
Table[Length[Cyclotomic[n, x]], {n, 1, 100}] (* Artur Jasinski, Jan 15 2007 *)

Prog

(PARI) a(n)=sum(k=0, eulerphi(n), if(polcoeff(polcyclo(n), k), 1, 0))
(PARI) a(n) = #select(x->x!=0, Vec(polcyclo(n))); \\ Michel Marcus, Mar 05 2017

Crossrefs

Cf. A086765 (number of positive terms in n-th cyclotomic polynomial), A086780 (number of negative terms in n-th cyclotomic polynomial), A086798 (number of zero terms in n-th cyclotomic polynomial), A087073.

Sequence in context: A307994 A238480 A111089 * A318884 A280990 A327667

Adjacent sequences: A051661 A051662 A051663 * A051665 A051666 A051667

Keyword

nonn

Author

Jud McCranie

Extensions

More terms from Labos Elemer, May 03 2002

Status

approved