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 A051664 a(n) is the number of nonzero coefficients in the n-th 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 (list; graph; refs; listen; history; text; internal format)
 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 EXTENSIONS More terms from Labos Elemer, May 03 2002 STATUS approved

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Last modified April 3 04:21 EDT 2020. Contains 333195 sequences. (Running on oeis4.)