A086765 -id:A086765 - cp's OEIS frontend

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

Jud McCranie

nonn

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

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

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

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.

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

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

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

a(n) = #select(x->x!=0, Vec(polcyclo(n))); \\ Michel Marcus, Mar 05 2017

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

More terms from Labos Elemer, May 03 2002

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

Original entry on oeis.org

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

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

nonn

See A051664

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

Cf. A086765, A000961.

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

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

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

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

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

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

Original entry on oeis.org

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

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

nonn

See A051664.

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

Cf. A086765, A086780.

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

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

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

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

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

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

A085459 Numbers k such that k-th cyclotomic polynomial has exactly 3 positive coefficients.

Original entry on oeis.org

3, 9, 10, 20, 27, 40, 50, 80, 81, 100, 160, 200, 243, 250, 320, 400, 500, 640, 729, 800, 1000, 1250, 1280, 1600, 2000, 2187, 2500, 2560, 3200, 4000, 5000, 5120, 6250, 6400, 6561, 8000, 10000, 10240, 12500, 12800, 16000, 19683, 20000, 20480, 25000, 25600

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 14 2003

Sequence appears to consist of 3^i, i > 0; and 2^i*5^j, i, j > 0. Are there any other terms? - David Wasserman, Feb 01 2005

			9 is a member because the 9th cyclotomic polynomial is P(x) = x^6+x^3+1.

Cf. A065119, A086765.

Select[Range@ 5000, Count[CoefficientList[Cyclotomic[#, x], x], ?(# > 0 &)] == 3 &] (* _Michael De Vlieger, Oct 26 2017 *)

n = 0; while (1, n++; p = polcyclo(n, x); d = poldegree(p); c = 0; i = 0; while (c < 4 && i <= d, if (polcoeff(p, i) > 0, c++); i++); if (c == 3, print(n))); \\ David Wasserman, Feb 01 2005

More terms from David Wasserman, Feb 01 2005

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

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

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

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A085459 Numbers k such that k-th cyclotomic polynomial has exactly 3 positive coefficients.

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