

A051664


a(n) is the number of nonzero coefficients in the nth cyclotomic polynomial.


14



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
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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) = 2xy1. 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. 979981.
T. Y. Lam and K. H. Leung, On the Cyclotomic Polynomial Phi(pq,x), Amer. Math. Monthly, Vol. 103, No. 7, 1996, pp. 562564.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial


FORMULA

a(n) = phi(n)+1A086798(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 nth cyclotomic polynomial), A086780 (number of negative terms in nth cyclotomic polynomial), A086798 (number of zero terms in nth cyclotomic polynomial), A087073.
Sequence in context: A162753 A238480 A111089 * A318884 A280990 A256267
Adjacent sequences: A051661 A051662 A051663 * A051665 A051666 A051667


KEYWORD

nonn


AUTHOR

Jud McCranie


EXTENSIONS

More terms from Labos Elemer, May 03 2002


STATUS

approved



